Aug., I898. ] AIt [Jzstructive 3fechamcal Failure. IO5

AN I N S T R U C T I V E M E C H A N I C A L F A I L U R E .

BY WILFRED LEwiS,

Member of the Institute.

The discovery of b road general principles is cons tant ly re- mov ing from the field of research the hal luc inat ions tha t former ly engrossed the act ivi t ies of many hands and minds, and no th ing i l lus t ra tes more clearly the va lue of technical t ra in ing than the check which it imposes upon the pursu i t of mechanica l follies. An example of this may be noted in the l ives was ted on the p rob lem of p e r p e t u a l mot ion tha t migh t have been saved for usefu l work by an earlier expo- si t ion of the conservat ion of energy. T h a t a few are still engaged in the hopeless task is clearly due to the imperfec t d isseminat ion of this g rea t pr inciple; bu t since the t ime of Redheffer , the n u m b e r of these misgu ided en thus ias t s has s teadi ly diminished.

The craze for perpe tua l mot ion p robab ly reached its he igh t in i812, when, as descr ibed by Dr. H e n r y Morton, in this Journal for April , 1896, Redhef fe r appl ied to the Legis- la ture of Pennsy lvan ia for a g ran t of funds to carry on and perfec t his g rea t invention. Ins tead of acceding to this modes t request , the more p ruden t course was adopted of appoin t ing an inves t iga t ing commit tee , which was gra- c iously al lowed to v iew the wonder of the age at a respect- ful distance, t h rough a glass case, b u t closer inspect ion was not invited. Never the less , one glance was sufficient for the keen observa t ion of y o u n g Coleman Sellers, ~ as the resul t of which a dupl ica te model was soon made and exhibi ted wi th ve ry depress ing effect upon the pre tens ions of Red- heifer, whe reupon he re t i red to pr iva te life.

Tfiere was no re fuge then for an unmasked deceiver beh ind the bars of a fantas t ic t e rminology known only to himself . Doubt less , people l iked to be h u m b u g g e d then as they do n o w ; b u t no gen ius had ever conceived the poss ib i l i ty of success in a series of occul t scenic effects

Afterwards the father of our esteemed eonte.mporary, Dr. Coleman Sellers.

I06 Lez c~ i s ." [J. F. I.,

pu rpor t ing to i l lus t ra te a confused m u m m e r y of terms. This t r iumph was reserved for a la ter period, when sa t ia ted reason seemed to crave diversion. I t is not m y intent ion, however , to reckon wi th the phi losophy of the Kee ley motor as expounded by its a l leged inventor, nor to deal wi th any th ing beyond the pale of common sense.

But, a l though perpe tua l mot ion has been re lega ted to the bo t tomless pi t of folly by all minds capable of g rasp ing the t ru th which it violates, there are o ther t ru ths equal ly well es tab l i shed as laws of nature , aga ins t which the folly of would-be inventors is still beat ing. I t is not surpr is ing tha t in I812, whi le the ph i losophy of ene rgy was in process of evolut ion and before the g rea t doctr ine of its conservat ion had been clearly es tabl ished, in tense in te res t should have been awakened in the r epu ted d iscovery of perpe tua l motion. There was then some hope of success and the promise of fame and for tune to the t r i u m p h a n t inventor . But, at the close of this cen tu ry of scientific progress, it is amazing to wi tness an assaul t upon pr inciples fo rmula t ed long before the conservat ion of energy, and clearly eluci- dated in modern text-books on na tura l phi losophy.

Iner t ia is an inheren t p roper ty of mat ter , and Newton ' s first law of motion, which is really a law of inert ia, asser ts tha t a body at res t remains at rest, and a body in mot ion

c o n t i n u e s in mot ion and in a s t ra igh t line, unless i t is deflected by some control l ing force.

Newton ' s laws also asser t the re la t ions be t ween force, mass and accelerat ion, and define force in t e rms of the accelerat ion p roduced upon a g iven mass in a giver, t ime. Phys ica l forces are thus made comparab le to the force of gravi ty , and the measure of force is expressed in accelerat ion or change of veloci ty. The ac tual ve loc i ty has no th ing to do wi th it except as an index to the ra te of change in deflected or curvi l inear motion, and, in every case, the change of veloci ty in any given direct ion is the t rue measure of the force act ing in tha t direction.

The laws of g rav i ty and inert ia have for centur ies been known to hold the p lanets in thei r orbits, and the observed effect of these laws on the mot ion of m a t t e r in space has

Aug., x898. ] 2tn Instructiz,e i~[echaltical Faihtre. xo7

served to detect the existence of unseen mat ter and point to its actual discovery, thus demonstrat ing the universality of natural law and its perfect precision of action in the boundless depths of space. Here the problems are grand and complex, while the data for their solution are often incomplete and uncertain, requir ing the exercise of rare judgment and great ability; but in dealing with terrestrial bodies, where, motion is necessarily more limited and restrained, the effect of inertia is felt directly in the restraining material, and the data for its determination are so definite and clear that no room is left for doubt or speculation of any kind. Yet on the 9th of March, x897, we find the U. S. Patent Office actually granting patents for an alleged improvement in balancing locomotive driving- wheels on the pretension that the translation of the wheel along the rail has an important bearing, hi therto overlooked, upon the inertia of the revolving parts. Such action totally ignores the well-known fact that the revolving parts never give any trouble in balancing, and present no difficulty whatever to be overcome, and naively assumes the existence of an imaginary fault for the purpose of having something to correct, while the real difficulty of balancing the recipro- cating and revolving parts together seems to be unheeded or unknown.

It would hardly be worth while to give an idea of this kind more tt]an passing notice, had it not been so persist- ently entertained and developed, and did it not appear to be gaining credence to a remarkable extent.

The amount of time, money and enthusiasm spent in this direction can be appreciated only after an examination of the apparatus designed to sustain the contention raised, and a review of the arguments and diagrams offered in its support.

All of this is so ingenious, so plausibly presented and throws such interesting and unexpected side lights upon a subject not commonly studied, that it is hoped its exposi- tion may compensate in some measure for the complete failure of the original purpose to improve the balance of locomotive driving-wheels.

IO8 L e w i s : [J, F. I..

The p romote r of this l audable scheme was un fo r tuna te ly misgu ided in his percept ion and in te rpre ta t ion of facts, b u t so confident was he of the success of his labors tha t he applied, not for a g ran t of funds f rom the Legislat t i re , like the over-reaching Redheffer , bu t for a repor t f rom an inves- t iga t ing commi t t ee of the Frankl in Ins t i tu te .

I t has been the pr ivi lege of the wr i te r to serve on this commit tee , and, wi th the gene rous approval of the appli- cant, who desires the t ru th to be known, hop ing tha t o thers m ay benefi t by his experience, the s tory of this fai lure in ba lanc ing may now be told.

The al leged improvemen t in ba lanc ing a locomot ive dr iving-wheel is shown and descr ibed in three U. S. pa ten t s

FIG. I .

gran ted to Phi l ip Z. Davis, March 9, I897, and n u m b e r e d consecut ive ly 5 7 8 , 5 9 7 - 8 - - 9 . In the f i rs t -ment ioned pa t en t a n u m b e r of we igh t s are disposed, as shown in P~g. i, four of which, inc luding the crank pin, are in the crank circle 9 °0 apart , while four more are in an ou te r circle 9 °o apart, and on radial l ines 45 ° f rom the inner set. The weights , C, in the crank circle, are descr ibed as be ing each equal to the we igh t of the boss D and the pin E, while the equal we igh t s B, in the ou te r circle, are said to be var ied in we igh t wi th the rat ios b e t w e e n the d iameters of the crank circle, the ba lane ing circle and the rollin K circle, bu t in wha t m a n n e r does not appear.

Aug., i898. ] AJI IJzstructive Mechanical Failure. IO 9

In this a r r a n g e m e n t there is but one we igh t opposed to the crank pin, all the o ther weights being made to balance each other.

~'I@. 2.

In the second patent , No. 578,598, there are two,weights opposed to the crank pin, as shown in Fig. 2, 120 ° apart, and this a r r a n g e m e n t is put forward as the most practical form of the invention.

~Z

F I G . 3.

In the th, ird patent , No. 578,599, there are, as shown in -~'~. 3, two balancing rings, one in the crank circle and one outside, and apparen t ly there is no th ing direct ly opposed as a coun te rwe igh t to the crank pin.

I IO L e z v i s ." [J. F. I.,

In all cases the wheels are a s sumed to be in runn ing balance on their centers, and to the pecul ia r a r r angemen t s of coun te rwe igh t s is ascr ibed the v i r tue of ma in ta in ing this balance while the whee ls are runn ing on a track.

I t cannot be denied tha t the wheels , ba lanced as de- scribed, will run in perfect ba lance on a s t r a igh t t rack jus t l ike any o ther ba lanced wheels; b u t the i m p u t a t i o n is tha t the usual me thod of ba lanc ing is defect ive, and adopt ing this fiction as an hypothes is , a cur ious ph i losophy is devel- oped to explain a shadow tha t was never cast.

The inventor claims a sys tem of ba lanc ing b y which he ob ta ins a per fec t ba lance for all revolv ing parts, and an im- proved balance for the rec iproca t ing par ts of a locomot ive driving-wheel, b u t his a rgumen t is all wi th reference to the revolv ing parts , and as no th ing is adduced to show any im- p rovemen t in the ba lance of the rec iproca t ing parts, our a t t en t ion will be confined to the revolv ing par ts only. H e also c la ims that , b y us ing two coun te rwe igh t s d isposed as shown in Fi K. 2, he has overcome the difficulty in ba lanc ing locomot ive drivers, and his ph i losophy asser ts tha t any we igh t a t t ached to a locomot ive dr iving-wheel m u s t have i ts force and effect compu ted from its t rue cen te r of rota- tion, which is the poin t of con tac t of whee l and rail for the mot ion of ro ta t ion and t rans la t ion combined .

To demons t r a t e in a pract ical way this fundamen ta l con- t en t ion tha t the mot ion of t rans la t ion cannot be neglec ted in compu t ing the effect of a coun te rwe igh t , and to show tha t a wheel ba lanced b y two we igh t s for ro ta t ion a round its center is not in ba lance w h , n the mot ion of ro ta t ion and t rans la t ion are combined b y roi l ing on a track, the t e s t i n g mach ine i l lus t ra ted in Fig. 4 was des igned and bui l t .

This machine consis ts of a circular track, abou t 3 fee t in diameter , upon which a pai r of coun t e rwe igh t ed discs is m o u n t e d to run on hor izontal axles, while the axles them- selves are dr iven b y a ver t ical shaf t in the cen te r of the cir. cular track. Th i s t rack is m o u n t e d on three we igh ing levers connec ted at the center to ano ther lever, which in turn is connec ted to the elast ic finger of a record ing pencil. The ver t ical shaft , which drives the pair of discs, carries a large

Aug., I898.1 A1z Dzs/ructive Alcc/mnical Failure. I I I

paper drum, upon which the recording pencil can be al lowed to act. The axles which carry the two discs are h inged near the center, leaving the discs perfect ly free to press upon the circular track, and any var ia t ion in this pressure is shown b y the m o v e m e n t of the recording pencil.

FIG. 4.

W h e n the discs are coun te rweigh ted , as shown in Fig. 4, and dr iven at a mode ra t e speed, the recording pencil will be se t in vibration, making two comple te m o v e m e n t s for every

x 12 L e w i s : [J. F. I.,

ro ta t ion of the disc, and, as the speed increases, the ampli- t u d e of the v ibra t ions becomes rapid ly greater . On the o ther hand, when the discs are coun tevweigh ted at th ree points, as shown ir Fig . 2, there is n o v ib ra t ion of the re- cording pencil at ny speed, b u t the posi t ion of the pencil changes, showing more and more pressure on the t rack as the speed increases.

The remarkab le resul ts indica ted b y this ingenious me- chanism have been supposed, by some observers , to subs tan- t ia te the content ion tha t the accepted ph i losophy of inertia, as commonly appl ied to coun t e rwe igh t s in ro t a t ing bodies , is all wrong when appl ied to bodies which combine a mo- t ion of ro ta t ion wi th tha t of t ransla t ion. In o the r words, " t h a t the force and effect of the coun t e rwe i gh t s m u s t be compu ted from their t rue center of rotat ion, which is the poin t of contac t of wheel and rail."

Th is is p u t forward as a new and o r ig ina lme thod , differ- ing sens ib ly in i ts resul t s and leading to a decided advance in the perfect ion of ba lanc ing a locomot ive driving-wheel.

The re is, however , an obv ious difference b e t w e e n the m o v e m e n t of the exper imenta l discs and tha t of a locomo- t ive driving, wheel, as it commonly occurs on a s t ra igh t track. The former ro ta te a b o u t an axis which is i tself re- vo lv ing abou t ano ther axis at r igh t angles, whi le the la t ter s imply rota tes abou t an axis in mot ion which remains par- a11el to itself. One gyra tes in two planes of motion, while the other moves only in one, and this impor t an t difference furnishes the key for expla in ing the observed facts as na tura l resul ts w i t h o u t the aid of the p re tended new theory of balancing.

I t cannot be admi t t ed as poss ib le tha t a wheel in runn ing balance on a fixed axis will be out of ba lance on an axis mov ing parallel to itself, nor tha t " t rans la t ion and ro ta t ion c o m b i n e d " on a s t ra igh t t rack can have any effect what- ever upon the ba lance of the revolv ing parts. T he effects observed in the exper imenta l appara tus are demons t rab ly due to gyroscopic action, and to show this in a pract ical w a y we are indeb ted to Mr. H u g o Bi lgram for the instru. men t i l lus t ra ted in Fzg. 5.

Aug., x898. ] An [tzstructiz,e 3[echaJlical Fail t trc. x 13

It is s imply a disc moun ted upon an axis in a forked handle, to be spun wi th a cord like a top, and held in the hand. Two or three coun te rwe igh t s m a y be a t tached to the disc, giving, in e i ther ease, a perfect runn ing balance.

Now, while the disc is spinning, it is found tha t the in- s t r u m e n t can be moved in any direct ion parallel to itself, wi th as much freedom as when the disc is at res t ; bu t at- t empt to change the direct ion of the axis, and a decided re- sistance is at once encountered.

If the disc is sp inning rapidly and the direction of the

I~IG" 5"

axis is suddenly changed, the i n s t rumen t twists in the hand even when a s t rong grip is taken to prevent it. This phe= nomenon occurs whe the r two or three counterweights are used, bu t a decided difference in effect is also noticed. W h e n two counterweights are used, the twis t ing referred to is accompanied by a series of impulses depending upon the speed of rotat ion, bu t when three are used the twist is s teady and wi thou t pulsations.

This exper iment is in te res t ing and instructive, not be- cause it exposes the fu t i l i ty of the elaborate tes t ing ma- VOL. CXLVI. No. 872. 8

I 14 L e w i s : [J. F. I.,

chine to establ ish the fundamen ta l content ion in this new philosophy, bu t because i t sugges ts the possibi l i ty tha t out of its fai lure and by its means a new and impor tan t princi- ple of mot ion may accidental ly have come to l ight. I t is not surpr is ing tha t the gyroscopic effect of three counter- weights should be s teadier than tha t of two, bu t it is at first surpr is ing tha t three weights should run as s teadi ly as four

F I G . 6 .

l Fro. 8.

_l

D

1 F I G . ~ #,

or six in two planes of mot ion ; and a r igid analysis of the problem discloses the curious fact, which may be s ta ted as a principle, tha t in gyroscopic effect three weights , disposed as shown in Fzg. ~, 120 ° apart, are equiva len t to a rin K of the same radius and weight . This m a y or may n o t b e a new discovery, bu t it leads to such a clear unders tand in K of the

Aug., 1898. ] 2t n [Jzstructive 3~[ec/tanical Faihtre. t 15

gyroscope, and explains so ful ly the observed effects in the t e s t ing mach ine exper iments , t ha t it is t h o u g h t advisable fo ~ v e the analysis in detail, a f ter which the fal lacy in the a r g u m e n t " f r o m poin t of con tac t with r a i l " will be con- sidered.

Given the disc D, Figs. 6, 7 and 8, radius r, moun ted on a hor izonta l axis and dr iven on the circular t rack T, radius R, by a ver t ica l shaf t C, normal to the plane of the t rack and pass ing t h r o u g h its center , the disc be ing free to press upon the track, bu t r es t ra ined by its axis, or by a coun te rwe igh t , aga ins t cen t r i fuga l force about the cen te r C.

(i) To find the effect on the t rack of two equal masses, MI and 3//.2, Fig. 7 at a d is tance r f rom the axis and I8o ° apart .

(2) To find the effect of th ree equal masses, M1, M2 and M3, Fig. 8, i2o ° apart . Le t v~ = ve loc i ty in feet per second of the cen te r of the disc

D abou t the ver t ica l axis C. v ~--- ve loc i ty of any o the r point in the same direct ion at a

d is tance x above the track. R 1 - - radius C D in feet at which the center of the disc D

revolves about 6". r I --= radius of the disc D in feet. r = radius of the masses M~, M s and 313 to be considered. w --~ angu la r veloci ty of the axis C D.

t~ = angu la r ve loc i ty of the disc D abou t its axis C D --~

CO - - - - ~'1 r l

at ~ accelera t ion normal to the plane of the disc D due to the veloci ty v .

a --~ accelera t ion normal to the plane of the disc D due to the veloci ty v.

F r o m the condi t ions of the prob lem it is ev iden t tha t the veloci ty v, of any poin t in the disc D at a d is tance ~ above the t rack T, may be expressed by the equa t ion

.~" 7/1 - - ( I )

T1

1 16 Le~vis : [J. F. I.,

T h e c e n t e r of the disc D m o v e s a b o u t C w i t h t h e v e l o c i t y vl = ~o R 1 a nd i ts n o r m a l a c c e l e r a t i o n is g i v e n in t h e well- k n o w n e x p r e s s i o n for c i r c u l a r mo t ion ,

a 1 = ~0 2 R 1 -~- ¢0 V 1 ( 2 )

H e r e al, ~o a nd Vl are t a k e n in a p l a n e n o r m a l to the ver t i - cal axis C, a nd s ince ~o is c o n s t a n t for all p o in t s in t h e disc D, t he n o r m a l a c c e l e r a t i o n a fo r a n y po in t in t h e disc m u s t d e p e n d u p o n i ts v e l o c i t y v and be e x p r e s s e d b y t h e g e n e r a l e q u a t i o n

X X a = Go v = ~o z, 1 - - = a I ( 3 )

f l f l

T h e fo rce Y d e v e l o p e d b y a mas s 2g u n d e r the acce lera- t i o n a is F = M a, b u t s ince t he b a l a n c e d m a s s e s are all a s s u m e d to be equa l , t h e y m a y be m o r e c o n v e n i e n t l y t r e a t e d as u n i t masses , fo r w h i c h we h a v e s i m p l y F = a.

A t t he c e n t e r of t he disc D t he n o r m a l fo rce F0 for a u n i t of ma s s a t t h a t po in t b e c o m e s

F0 = a: = eo vt (4)

and for a n y o t h e r p o i n t in t he disc a t t he d i s t a n c e x a b o v e t h e t rack ,

F : ~ F0 (5) r l

I n t he case s h o w n b y Fig. 7, two equal m asse s M~ a n d M., a re a s s u m e d at t he d i s t a n c e r f r o m the c e n t e r of t h e disc D an d 18o ° apar t .

For M: w e h a v e x = rl + r and Fl = ( , + ~ ) Fo

F o r M s we h a v e x : r l - - r a n d F.2 : ( I - ; ; ) F 0

T h e m o m e n t K1, of t he force F: a t t h e d i s t a n c e r a b o v e t he d r i v i n g t r u n n i o n s C is

Kl = r Fl : r ( I + r ) Fo rl

an d the o p p o s i n g m o m e n t of the force F., is

Aug., I898. ] /tJl Ins t ruc t ive Mechan ica l Fai lure . I 17

& = - - r F 2 = - - r I ---7.1

T h e Mgebraical sum of these moments , or the resu l t an t m o m e n t K, is the re fore

K = KI _b K 2 2r2 - - F 0 : 2 r 2 ~o fl ( 6 )

Now, when the masses M1 and M'~ turn t h r o u g h 9 °0 and s tand for an ins tan t at the same he i gh t r I above the t rack 27, it is ev iden t t ha t the r e su l t an t m o m e n t K is zero, and tha t as the disc D runs on its c i rcular track, K mus t va ry by the a m o u n t 2 r ~ w fl twice for each ro ta t ion of the disc. Div id ing the m o m e n t 2 r ~ ~o fl by the a r m R 1 t h r ough which it acts, the var ia t ion in the pressure on the t rack may be expressed by the equa t ion

a l p _ _ 2 r 2 , o f f (7 )

R1

Since the cen te r of g rav i ty of the masses M1 and M2 is a lways in the cen te r of the disc D, it is ev iden t tha t the sum of the forces F1 + F~ = z/7o is constant , and deduc t ing F0 f rom F1 and F.z respect ively , we have the componen t s F~ - - F0 and/~'2 - - Fo caus ing m o m e n t s abou t C, and the cen- tral force z F0 a long the radius C D.

Fl "Fo = r Fo a n d F2 _ Fo _ r Fo ~1 ¢'1

the first of which acts at the d is tance r, caus ing the m o m e n t

- F 0

while the second acts at the d is tance - - r, caus ing the same moment , the sum of which is

r 2

2 F o = 2 r~ w fl r l

as g iven in equa t ion (6). For any a r r a n g e m e n t of equal ba lanced masses, the sum

of the m o m e n t s K can the re fore be expressed as the sum of the componen t s y~ ~o fl, in which y is the ver t ical dis tance of

I I 8 L e w i s ." [J. F. I.,

each po in t of mass above or be low the p lane of m o t i o n for the axis C D.

In o the r words , we h a v e

K = s y ' ,o (8)

b y w h i c h the ana lys i s for th ree or more m a s s e s can be eas i ly fo l lowed.

I n F @ 3 t he re are t h r ee u n i t masses , M1, M2 and M~, a t the d i s t ance r f rom the cen t e r a n d izo ° apar t . _/g~ acts at t i le d i s t ance r above the center , M:.~ and M3 each act a t the d i s t ance r sin. 3 °° be low the center , and the s u m of the

m o m e n t s K is, b y e q u a t i o n (8),

K = (y2 + y Z + ys~) ~o p or

K = , o 3 = i . 5 r (9)

Now, if we let t hese masses keep the s a m e r e l a t i v e posi- t ions a n d be s h i f t e d t h r o u g h an ang le 0, we h a v e

Yl = r cos. O, Y2 -= r sin. (3 o° O) and Y3 = r sin. (3 o° -t- O)

y , = r ~ cos3 O, y ] = r2 s in3 (3 o° - - O) and y ] = r 2 sin3 (30 ° _t_ O)

sin. ~ (30 ° - - O) = ¼ cos3 0 - - 1 / ' ~ s i n . 0 cos. 0 + "75 s in3 0

s in3 (3 o° + 0) = ¼ cos3 0 + t / ' 7 5 sin. O cos. 0 + 75 sin3 0

T h e r e f o r e y t 2 -1- y ] -1- y 2 = i. 5 r ~ ( s i n ? 0 -~- cos . 2 O) w ~ =

I'5 r2 o) i9, as s h o w n by e q u a t i o n (9) for the o r ig ina l posi t ion.

W i t h th ree u n i t masses a t the s ame d i s t ance f rom the cen t e r and I2o ° apar t , t he re is, the re fore , no c h a n g e in the r e s u l t a n t m o m e n t of the c e n t r i f u g a l forces, a n d for the con- s t a n t p ressure on the t rack due to these forces, we h a v e

p _ _ l " S r 2 w ~ (IO) RL

Simi l a r ly i t m a y be s h o w n t h a t the to ta l enerffy E s to red in t he masses Mr, Y/.2 and M a is cons t an t , and for th i s the va lue is expressed b y the e q u a t i o n

E = 3 ~°2 R, + 3 ~2 r + 1"5 (o ~ r (I I)

T h e s u m of the c e n t r i f u g a l forces F =- F~ + F2 q- F3 is

Aug., i898. ] AJz [Jzstructive Mechanical Faihtre. I 19

also cons tant and equal to 3 P0, and it mus t therefore be concluded tha t three equal masses I2O ° apart, as shown in Figs. 2 and g, will run as s teadi ly as a homogeneous r ing wi thou t var ia t ion in pressure on the track or periodic im- pulses of any kind.

In the above analysis, the disc D has been t reated as a very th in sheet of metal, and the masses M1, Ms and M~ as points of weight in tha t plane, whereas the disc mus t have sensible thickness and the counte rweigh ts mus t have volume.

From the results obtained, however, the analysis is easily ex tended to a disc of any thickness and to weights of any size as cyl inders therein,

Since three points of mass, as shown, are equivalent to a homogeneous r ing of their combined mass at the same radius, it can be shown from equat ion (9 ) tha t a r ing of un i t mass at the radius r will exert the m o m c n t

2~'r ~ - "5 r2 ,0 fl ( I 2 )

and from this equat ion it is evident tha t the momen t f r is independen t of the posit ion of the assumed r ing along the axis C D. Equat ion (i2) therefore applies to a th in cyl inder of any length, and from this the momen t K a for a solid disc of uni t mass will be found to be

= "25 r ,o p

For three cyl indrical counterweights , each of un i t mass, 12o ° apart, hav ing a radius ro and ac t ing at the radius r from the axis C D, we have

f 3 = I '5 g 2 , 0 t~ --~ "75 %' w fl (I4)

I t is thus possible to de termine for any given disc, weighted as shown in Figs. 2 and 8, the increase in t rack pressure due to its speed and the curva ture of the track. Obviously, if the t rack is s t ra ight , w becomes zero, and there is no increase of pressure.

Refe r r ing again to Fig. 7, it should be observed that , a l though the sum of the cent i fugal forces F 1 + F2 is con- s tan t and equal to 2 Fo, the total energy of mot ion E, like the track pressure, is variable. For two uni t masses at the

120 L e z v i s ." [J. F. I.,

r ad iu s r, th i s v a r i a t i o n in e n e r g y a E m a y be e x p r e s s e d b y t he e q u a t i o n

A E = 2 r 2 co" (I.5)

and th i s will se t up p lus and m i n u s i m p u l s e s in the p l an e of the axis C D, pa ra l l e l to t he t rack .

A disc w i t h two b a l a n c e d masses , 18o ° apar t , is t h e r e f o r e c h a r a c t e r i z e d wh i l e r u n n i n g on a c u r v e d t rack , b y i m p u l s e s u p o n a nd a l o n g t he t rack , t he l a t t e r t e n d i n g to f l u c t u a t e t h e a n g u l a r ve loc i t i e s co a n d fl, wh i l e the f o r m e r s i m p l y v a r i e s t he p r e s s u r e on t he rail.

F o r c o n v e n i e n c e in e s t i m a t i n g th i s v a r i a t i o n in ra l l pres- sure , e q u a t i o n (7) m a y be w r i t t e n in t he f o r m

A P - - 2 r 2Co,,5 (I6) G

w h e r e G, t he g a u g e of t he t rack , b e c o m e s t h e a r m of t h e couple .

F o r e x a m p l e , s u p p o s e a d r i v i n g - w h e e l 6 f e e t in d i a m e t e r ha s a mass of Io c o n c e n t r a t e d a t t he c r a n k p in I foo t r ad ius , and an e q u a l mas s a t t he s a m e r a d i u s d i r e c t l y oppos i t e , an d le t th i s w h e e l be r u n n i n g on a c u r v e of i ,ooo f ee t r a d i u s a t t h e r a t e of 9 ° f ee t a second , or a l i t t l e o v e r a m i l e a m i n u t e , the g a u g e of t h e t r a c k G b e i n g 5 feet , to find t h e v a r i a t i o n in ra i l p r e s su r e .

H e r e co- - 90 - - '09, a n d i ~ - - 90 - - 3 ° . S u b s t i t u t i n g iooo 3

these v a l u e s in e q u a t i o n (i8), we h a v e

A P = 2 X Io X "o9 X 3 o = 8 . 4 1 b s . 5

a v e r y i n s ign i f i can t a m o u n t for q u i t e an e x t r e m e ca se ; b u t th i s r e f e r s to one d r i v e r only , wh i l e the o t h e r d r i v e r on t h e s a m e axle m u s t h a v e t he s a m e m a s s e s d i s p o s e d a t r i g h t ang le s to t he first.

T h e r e are, c o n s e q u e n t l y , f ou r e q u a l m a s s e s 9 °0 a p a r t to be cons ide red , a n d r e f e r r i n g to t he g e n e r a l e q u a t i o n for t h e m o m e n t of c e n t r i f u g a l forces , K = E y2 co fl, we h av e , w h e n :If 1 a nd M.~ are in a ve r t i c a l l ine and M2 and Af~ in a h o r i z o n t a l

Aug., i898. ] All flzstrttcth,e JFZec/zaltical F a i l u r e . 121

line, at the radius r, yl = r, y2 = o,y~ = r andy4 = o. T h e sum of the m o m e n t s y2 w/~ is, therefore ,

K ~-~ 2r 2 co i~ (I 7)

Now, le t t ing all these masses tu rn t h rough a small angle O, we have

Y l = r cos. O, y2 ~ r s in . O, y~ ~ r cos. O, y4 ~ r s in . 0

and the sum of the m o m e n t s becomes

K = 2 r 2 ( s i n ? 0 + cos? O) ,o i~ = 2 , 2 ,o

as before. If the crank pins are ba lanced by equal masses d i rec t ly opposite , it thus appears tha t no f luc tua t ion in t rack pressure can occur f rom this cause even when the wheels are r u n n i n g on a curved t rack at h igh velocity. I t is also qui te apparent , by the m e t h o d jus t employed, tha t this conclusion applies as well to a pair of dr ivers counter- we igh ted in the usual manne r by we igh t s near the r im in- s tead of in the crank circle, and it cannot be doub ted tha t the usual m e t h o d of ba lanc ing a pai r of locomot ive driving- wheels is as per fec t in effect as any of the me thods claimed to be an i m p r o v e m e n t in the pa ten t s re fe r red to.

T h e recognized difficulty in coun te rba l anc ing driving- wheels, on accoun t of the c o u n t e r w e i g h t being in a di f ferent plane of mot ion from tha t of the crank pin, and also on account of the iner t ia of the r ec ip roca t ing parts, r emains unnot iced , and need not be discussed. W e now come to the a r g u m e n t " f rom poin t of con tac t with rail," in suppor t of which the gyroscope has been so fu t i le ly employed.

I t is c la imed tha t the force and effect of a c o u n t e r w e i g h t in a locomot ive dr iv ing-wheel m us t be compu ted f rom its t rue cen te r of ro ta t ion, the poin t of con tac t of wheel and rail for the mot ion of ro ta t ion and t rans la t ion combined.

Since iner t ia forces in a g iven di rec t ion are invar iab ly accompanied by a change of ve loc i ty in tha t direct ion, it is impossible to imag ine how the accelera t ions and retarda- t ions due to ro ta t ion can be at all affected by rec t i l inear t r ans la t ion in any direct ion.

T h e addi t ion of a cons tan t to a var iab le never affects its differential , and it is c lear ly the difference in veloci ty

I22 Lewis: [J. F. I.,

d iv ided by the difference in t ime tha t measures accelera- tions. This, in fact, is the definition of the t e r m ; b u t to show more conclus ive ly the fal lacy in the a rgumen t pre- sented, and the u t t e r lack of any foundat ion w h a t e v e r upon which the al leged i m p r o v e m e n t in ba lanc ing can be based, we will consider the ac tua l pa th t raced by a po in t in the dr iving-wheel wi th reference to the track.

The pa th t raced by any point b e t w e e n the center and the c i rcumference of a rol l ing circle is known as a prolate cy- cloid. A point in the crank circle G G, Fig. 9, t races the curve C G K, Fig. io, and a po in t in the ou te r circle H H, Fig. 9, t races the curve B IH , F,g. xo, also shown to a larger scale as B, B p, B 'r, etc., Fig. i i . These curves are bo th c o n -

"'ix." : /' '"'t • . - ' @ - ' - 4 .'" •

FIG. 9.

cave and convex to the track, and their poil~ts o f inflection, G G and H H , where the cu rva tu re changes, have been par- t icular ly specified in the f irs t-mentioned pa ten t as points where there is no normal force, the a r g u m e n t be ing tha t while the curve is concave, the cent r i fugal force in the mov- ing we igh t is away f rom the t rack ; and while convex, the centr i fugal force has a d o w n w a r d componen t upon the track. This would, undoubted ly , be t rue for a body moving in a prola te cycloid at a un i form rate, b u t i t is cer ta in ly not t rue for the var iable ra te at which the body ac tua l ly does move, because the inert ia of the body in the line of i ts t ravel is whol ly neglected.

Aug., i898.] AJz [JzstructiT,e 34recha~zical Failure. x23

No ma t t e r whe the r the center of the roll ing disc or its point of contact wi th the rail be considered as fixed, it is per- fect ly clear tha t any point in the disc a t ta ins i ts m a x i m u m vert ical veloci ty as it passes the center line, and that, conse- quent ly , on e i ther assumpt ion , the neu t ra l poin t in the ver- tical forces mus t be on that line.

FIG. IO.

Now, if the disc is ro ta t ing abou t its center O, Fig. zz, at the angula r veloci ty ~, we would cer ta in ly find the vert ical force exer ted by a uni t of mass at the poin t B, radius r by the equa t ion

y = p~ r ( I s )

A ~ A w A ~ *

FIG. i!.

and for the point B'"" we would have the same force in the opposite direction, but it is claimed that these forces should be computed from "the true center of rotation, the point of contact of wheel and rail."

Very well; let us look at the problem from the same point of view and accept without question the demonstration given in " Proctor's Geometry of Cycloids," that the radius

12 4 L e w i s ;

the prolate cyeloid

[ J . F . I . ,

the poin t B is ex-

F - - /~2 (~1 - - r ) 2 _ 192 r (2 5 ) ( r I - - r) 2

7"

both of which values of F are ident ical wi th tha t found by the usual me thod and expressed in equa t ion (I8).

I t thus appears tha t the cent r i fugal force at the points B and B " ' r of the prola te cycloid, Fig. zo, is the same, whe the r e s t ima ted from the center of the circle or f rom the poin t of contac t of wheel and rail, and, since the t rans la t ion of the wheel a long the rail means no th ing more than a cons tan t added to the hor izontal ve loci ty at any point as de te rmined for a fixed center, it is per fec t ly clear tha t t rans la t ion along the rail can not affect accelerat ions in any direction.

and s imilar ly for (20) and (22),

of curvature of at pressed by the equation

P - - ( r ' - ? r) 2 ( i9 ) r

and for the poin t B 't ' ' ' by the equa t ion

,o = ( r , - - r) 2 (20)

The veloci ty at the poin t B is clearly

7' = ~ (*'1 "~- r ) (21)

and at the poin t B ' " " it is

72 = f l ( r 1 - - r ) (2 2)

A uni t of mass mov ing wi th the veloci ty v at the radius p develops the eentr i fugal force expressed by the equa t ion

F - - (23) P

and subs t i t u t i ng in this equa t ion the va lues of p and z, for the poin t B, as g iven in equa t ions (I9) and (2o), we have

F - - (rl -t- r) 2 __/~2 r (24) ( r I -~- r ) 2

the point B " ' " we have, from equa t ions

Aug. , I898.] Alz fllstructiz,e ]~echauical Faihtre. I2 5

No fur the r demons t ra t ion is required, bu t it may be of interest to add a more general solution on geometr ic lines for the force of iner t ia developed at any point in a cy- Cloidal path.

Refe r r ing to F @ z2, let P be any point in the cycloid /3 A D' formed by the point A, in the rol l ing circle A C g on the b a s e D B D r . A C g is the axis of th.e cycloid, and the point A moves to P, when the point B" rolls to B r. The angle A p C I P is, therefore, equal to B C B ' , and twice the angle A p g ~ P.

Now, if the descr ib ing circle A C/] rolls at a uniform rate i t t u r n s t h rough equal angles in equal times, and since the angle A pB I P = 0 i s a l w a y s half of the angle B C B " , t h e line B t P, jo in ing the contact point B t with the describing point P, mus t also be moving with a uni form angula r ve- locity. Us ing our original notat ion, where rl = radius of the

a /

I

/~ 23' FIG. I2.

roll ing circle and ~tits angula r velocity, the velocity vl at the point A becomes vl = 2rl p, and the veloci ty v at any point P along the cyeloidal path is expressed by the equat ion

v = 2r l ~ cos. 0 (26)

Having found v, the accelerat ion a can be de te rmined from the general equat ion

d v a -- d t (27)

Different iat ing, equat ion (26)2gives d/9 and sub- d v -~ 2rl ~ ( - - sin. O) d O, in which we have d 0 =

2

s t i t u t i ng this value,

126 L e w i s . [J. F. I.,

d = - - r , u s in . 0 ( 2 s )

Obviously d t - - d/~

and we have a d v - - - - rx 32 sin. 0 (29) d t

in which the minus sign indicates re ta rda t ion . To find the certtr ifugal force at the point P ac t ing in the

direction B ' P , we have the veloci ty z, by equa t ion (26), and the radius of curva tu re at P being known to be 2 Bt P -- 4 r~ cos. O, the cent r i fugal force for a un i t of mass is given by subs t i t u t i ng these values of v and R in the genera l formula

(2 rx cos. 0) 5 F = R, whence F = 4 rt cos. 0 = rl cos. 0 (30)

We now have a mass at P under the impulse of two forces, a and F, as expressed by equat ions (29) and (30). The acceleration a is proport ional to the sine of the angle 0, and the cent r i fugal force F is proport ional to the cosine of the same angle. These forces may therefore be represented by the lines A t P and B' P, the resu l tan t of which is a d iameter of the roil ing circle in the direct ion P 6". Or, from equa- t ions (29) and (3o), we have for the resu l tan t radial force

S -- 1/a = q- ,~ = rl t~ 2 (3I)

which will be recognized as the general equat ion for centri- fugal force in a un i t of mass moving about a fixed center at the radius rl with the angular velocity ~.

The pa ten ts referred to in w h a t has preceded are thus shown to be in contempt of well-established principles, and therefore worthless.

T h e y are as clearly un tenab le as the idea of per- pe tua l motion, and ye t they are the ou tg rowth of hones t enterprise and toil, d i l igent ly cont inued for m a n y years. Technical t ra in ing in the r igh t direct ion migh t have saved the menta l energy a n d mater ia l subs tance so lavishly th rown away. The moral, " A lit t le learn ing is a dangerous th ing," remains as a warning, and as a f rui t ful result, per- haps some in teres t may be awakened in a mys ter ious toy of considerable scientific importance.