campbell-a periodic solution for a certain problem in mechanics

8/9/2019 Campbell-A Periodic Solution for a Certain Problem in Mechanics

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A Periodic Solution for a Certain Problem in MechanicsAuthor(s): J. W. CampbellSource: The American Mathematical Monthly, Vol. 34, No. 4 (Apr., 1927), pp. 188-195Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2299863

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2/9

188 A PERIODIC SOLUTION FOR A CERTAIN PROBLEM IN MECHANICS [Apr.,THEOREM IV. The square of the standarddeviation, OL, of a Lexis series,is givenbythe ormula:

(n -rnaL = npq+ ( ) (pi-p)2, where = (p+p2+ .+ +pr)/r and q =1-p.r i=1

PROOF: For thecalculation f thesquareof thestandarddeviation,weconsider he expression:oL2 = [fo(np - 0)2 + fl(np - 1)2+ ?+ f(np - n)2]/rN

= [n2p2(fo f i * **+ fn)/rN] [2np(0 fo+ 1 .f+ ?+ nfn)/rN]+ [(02fo + 12fi + 22f2 + + n2f,) /rN]

-n2p2 + Z (02fo(i)+ 12f,(i) + 22f2(i) + . + n2f i)) rN.From hefact hat

nE f/i)(np, - j)2 = Nnpiq,i=0we have, after quaring nd collecting erms,

02fo(i) + 12fl(i) + 22f2(i) + + n2f,(i) = Nnpiqi + Nn2p 2Hencewe getr r r[02fo(i) 12fl(i) + + n2f,(i)]/r Al nptqi + n2 p2 )/rr 2_

Pi n2E Zp2 f?pn2ni(i ]/=pi)2=1 r i=ASubstitutinghisvalue above, we have the formulan2_ n r

cL2 =-_n2p +np + E pi2r i=1which s easily een to be equivalent othe expressioniven nthe theorem.A PERIODIC SOLUTION FOR A CERTAIN PROBLEM INMECHANICS'

By J. W. CAMPBELL, University f AlbertaSupposea mass mrestingn a smoothhorizontalable, s connectedyalight nextensibletring, hich asses through smallhole n the centre fthetable,to a mass M hanging reely nd suppose hatin s given blow n a di-1Presented o the AmericanMathematicalSociety, ept. 9, 1926.


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1927] A PERIODIC SOLUTION FOR A CERTAIN PROBLEM IN MECHANICS 189rection t rightngles o the string n the table. Then f thecentrifugalorceexerted ymis equal to theweight f M, mwill revolve bout the hole n acircle nd M will remainn stationary quilibrium. f theblow mparted om is less than this critical alue thenthecentrifugalorce xerted ymwillbe less thantheweight fM andM willfall. But as M falls hedistance rommto thehole willdecrease, hecentrifugalorce xertedwill ncrease, ecom-ing equal to and then xceedingheweight fM. Thedownwardmotion f Mwillthusattain a maximum alue and thendecrease o zero,afterwhichMwill rise. The opposite ffect ouldbe obtained f theblow mparted o mweregreater hanthecritical alue. ThereforengeneralM willoscillatenavertical ine.'The above physicalphenomonon orms he basis of problems iven nLamb's Dynamics, page 148, and in Byerly'sGeneralisedCoordinates, age 22.Theproblemss there tated reonlyto find he totalrange fthe motion fM, and not to find general olution fthemotion, ut a generalperiodicsolution s here btainedwhen hetotalrange fmotionsnottoogreat. Themethod fobtaining he solutions analogous othemethods sed nthefind-ing of periodic rbits2n mathematicalstronomy,nd the mainpurpose fthearticle s to illustrate he applicability f those methods o problems fphysics ndmechanicswhere eriodic ariation s involved.The solution lsoshows omepeculiarities otcommon o astronomicalroblems.Let us denoteby r the distance fm from hehole,byx thedistance fMbelow heposition romwhicht starts, y0 theangular osition f thestringon thetable, ndbyT thetension f the string t time .Thenby the aws of motionwe obtainthe equations

m(r - r02) + T = 0m(r6+ 2M) = 0M - Mg + T = 0.Integratinghe secondoftheseequations nd eliminating between hefirst nd thirdwe obtain

r2o = Cl; m(r0-r2) - Mt + Mg = 0.We nowputr= a - x, and then, liminating between hese quations,weobtain

x(m + M)+ mc~1/(a x)3 - Mg = 0.1 It is assumedhat he ength f he tringnd the nitialonditionsre uch hat either norMreach heholebeforehe xtremeositionsfM arereached.2 F. R. Moultonnd collaborators,eriodic rbits,arnegie ublication61.


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190 A PERIODIC SOLUTION FOR A CERTAIN PROBLEM IN MECHANICS [Apr.,This equationmaybe writtenn theform

x + A/(a -X)3 - g,= O(1) A = mc?/(m + M), g, = Mg/(rn M).It follows rom heseequations hat. = 0 whenA/ a-x)3 =gl, that s whenx=a-(A/gi)"'3.Then put(2) x = a - (Al/gl)" + z,and the equationofmotion ecomesAz (Algl)113 Z)1}3 = lor Z + gi(l - k)-3 = gl, wherek3- g/A.

Expanding y the binomial heoremhis ast equationmay be writtenntheform ( ~~10z+ 3g,k1z + 2kz2 - k2z3+ 5k3z4- 7k4z5 = 0provided < Ilk. The k of thisequation s a parameterf theproblem ut tisnota convenientarametern which oexpresshesolutionwithout eneral-isation.To obtain periodicolutionwe makethetransformations(3) z = cy ; 3g,k= K; ck = X t -t = T (I + 3)/K} /2where 1 is the timeat whichx =a-lk, or at whichz=0 and at whichdx/dtis positive.The differentialquationthen becomes(4) y" + (1 + a){y + 2Xy2+10/3x2y3 + .. 0where heprimes enotedifferentiationithrespect o T.TheX s now taken s theparameterf the solution.The a is also a para-meterwhich s to be determinedo that the motion hall be periodicwithperiod r in T, the ntroductionf theK as abovemaking hispossible. Theintroductionfthec is tofacilitatehe discussionnd the constructionfthesolution,s thec maybe determinedo thaty'(0) = 1.For dtyI = z'/c= -/c (1 + 8)/K} 1/2/c,dTsince z = cy. Also by (2) c , and the energy ntegralof (1) givesA

+ - glx= c22 2(a -x)2


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1927] A PERIODIC SOLUTION FOR A CERTAIN PROBLEM IN MECHANICS 191At x=O, x=O and therefore2=A/2a2. Then at x=a-1/k we have

x2-A ( k2) + 2g, a- -Thereforefy'(O)= 1,

(5) c = { A -k2) + 2g1(a - i)} {2(1 + 3)/K} 1/2.It willbe convenientor ateruseto denote hefirstactorfthis xpressionbya single ymbol,nd accordingly e put

(6) y = {A (2 - k2) + 2g1(a )}Nowequation 4) maybe regardeds a differentialquation n two para-meters, andX, ndbythegeneral heoryfanalytic ifferentialquationstmaybe ntegrateds a power eries n andX, onvergingor andXsufficientlysmall or nypreassignednterval f ime. t is true hat he appearsmplicitlyin Xbut weshallgeneralisetandfor hepurpose fconstructionf he olutionweshallregardt as a constantwheret appears mplicitly.Let thepower eries olution n a and Xbe representedy00Y = E yiwviXisj=O

yxi(O) (i,j=O,* ,%(7) yOO(0)= 1

yiti(O)=O (i,j = O ... * ; i+j O).Substituting7) in (4) and equatingcoefficientsf 8iXj(i,j=O*,... oo)we

obtaina system fequationswhichmaybe integratedequentially.Thecoefficientf60X0 ivesyoo'+y0o 0; and yoo sinT s the solutionsatisfy-ingtheinitialconditionstated.The coefficientf 8X?gives y1'' +y,o=-sinr.The solution f thisequationsatisfying he initialconditionss readilyfound o be 1 TYio =--sin T+ -cosr.2 2Thesetwo teps re sufficiento establish he xistence fa periodicolutionof (4). Forthe conditions fsucha solutionwithperiod r in r are

(8) y(27r) y(O)= 0; y'(27r) y'(O) 1.


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192 A PERIODIC SOLUTION FOR A CERTAIN PROBLEM IN MECHANICS [Apr.,These conditions re not independent owever, s the firstmpliesthesecond. Forsincey(O) 0 and y'(O) 1, theenergyntegral f 4) implies hatif y(2w7r)=O,hen

y'(27r)} 2-{y'(0)} 2 = 1, andthereforey'(2ir) + 1.But sinceby thepower eries olutionust obtained

y=sinr +(- sinr + - cos r)6 +it follows hat(9) y(2r) =r6 + q(Q,X), q(0,0) = 0,(10) y'(27r) 1 + p(6,X), p(O,O)= 0.Equation (9) may be solvedfor6 as a power eries n X, vanishing ithXand convergingorX sufficientlymall. Also (10) is true forX sufficientlysmall. But whenX=0, by (10) y'(27r)1. Therefore '(2wr) +1, and not-1,forXsufficientlymall. Thereforehefirst f equations 8) implies hesecond.Again changinghe signs f y,X and r leavesthe differentialquation 4)unchangednd alsothe nitial onditions(O)= 0, y'(O)= 1 unchanged.There-forefy=f(X,6; r) is a solution, y=f(-X, 6; -r).

And f (X,6; r) is periodicwithperiod wx,hen (- X, 6; - r) is alsoperi-odicwithperiod2wr.Hence if6 is determinedo thatf(X, 6; 27r) 0, thenf(- X, ; - 2r) = f -, 6 ; 27) = 0.

Thismeans hat (X,6; 27r)s explicitlyven n X and thereforeis expan-sibleas a power eries n evenpowers f X.Let such an expansion e representedy(11) 8 = 82X2+ 84X4 + 66X6+ *

Nowby (3), (5) and (6), X2 c2k2 2k2(1+ 6)/Kand if we put , = y2k2/K,hen(12) 6 = 62A(1 6) + 64/u2(1 6)2 + 36A(1 + 6)3 + * - .

But this s againan implicit quation n 6and, whichmaybe solvedforas a power eries nA,vanishing ith andconvergingor sufficientlymall.The solutions6 = 32 + (32 + 64)Q2 + (623+ 36264+ 68)A3

With hisvaluefor6 thesolution7) is periodic.


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1927] A PERIODIC SOLUTION FOR A CERTAIN PROBLEM IN MECHANICS 193DirectConstructionf theSolution

Instead of findingf(X,3; r) in the manner indicated and solvingf(X, 3; 2-r) 0 for 3 to get equation (11), it is simpler to solve equation (4)directly s a powerseries nX, determininghe coefficientsf theexpansion (11)at each step so that the solution shall be periodicwith period2wrn ir.Accordinglywe assume the solution00 00y= I:yixi ; 6 = Y,62iX2ii-O i=l

yi(O) = O (i= o, * * 0)I V,(0) = 1, y '(0) 0 (i-1, * , ).

This solutionmust satisfy he equation identically n X,and thereforefterthe substitutionwe may equate the coefficientsfXiseparatelyto zero. Thereresultsa systemof differentialquationswhichmay be integrated equentially,the constantsof integrationbeingdetermined t each step so as to satisfytheabove conditions.ThecoefficientfXO ives yo' + yo= 0. The solution s yo sin.The coefficientf X gives yl' + y = - 2yo2 - 1 + cos 2ir.The solutionof this equation is 4 1y = + - cosr - - cos 2r.3 3ThecoefficientfX2gives

Y21 + Y2 -(4yoyl+ y03+ 2YO)/3/ 5\ 8 3

-a- 2 + - sinr - -sin 2r +- sin3r.For periodicity, 2= 5/6, and the solution of the equation gives

175 8 3Y2 = -- sin r + - sin 2r - - sin3r.144 9 16The coefficientfX3gives

/ ~ ~~~~~~5 \Ya" + y3 - 2y12 + 4YOY2 + 10yo2y + 5Yo4 - yi + y2)5 44 35= - - - cos 2r+ 6 cos3r - - cos 4r.6 9 18


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194 A PERIODIC SOLUTION FOR A CERTAIN PROBLEM IN MECHANICS [Apr.,The solutionof this equation is

5 199 44 3 7Y3 = - - --cos r + -cos 2r - -cos3r + -cos 4r.6 108 27 4 54This methodmaybe carriedon indefinitely nd as manyof the coefficientsas desiredobtained. The coefficientsf odd powers of X are cosine series inmultiplesfrandthecoefficientsfevenpowers re sine eries. t seems atherremarkablehatwhile hedeterminationf 32i is necessarynorder o elimi-nate the term in sin r in obtaining the coefficients f even powers of X, inobtaining the coefficients f odd powers the term in cos r drops out auto-matically.There eems obe no simpledirect roof fthisfact, ut that t is

true ollows rom he existence ndtheuniqueness f theperiodicolution.Also,the valuesof thecoefficientsbtainedmaybe checked ymeansofthe energyntegral fequation 4). That integral,fter implification,s4 5 7y'2 + a constant= - -(1 + -Xyl + _X2y4+ 2X3y5 _X4y6+3 3 3 /

This equation must be satisfied dentically n X, and therefore he substi-tution n it ofthesolution 12) requiresthatthefollowing elationsbe satisfied.yI2 + a constant- yo2/ 4 \2y Y 2yoyl + yc?

+2YJy2 3yl2 + Syoty2t _ (y2 + 2YOY2 + 4y Iy{ + - yo0+ +2y0{ ~~~~~~~~202yly + 2YY3 = - 2YY2 + 2YOY3 + 4y2Y2 + 4y0y 2 + 3yoi

+ 2yO5+ 52(2yoy1 + - yo0)}

Finally, hesolution n terms f the original ata is, on substitution,x = a-- + c{ sin -- (3-4 cos - + cos2-)X

- -- (175 sin -r 128 sin 2r + 27 sin 3r)X21441+ -8 (90 - 199 os T+ 176cos2T- 81cos3r+ 14cos 4r)X3+108


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1927] SUGGESTED READINGS IN THE THEORY OF NUMBERS 195where = (t - tl)/{ (1 + a)/K} 1/2 6 = 5/6g+

c2 = y2(1+ )/K; X = ck ; y2k2/KI 1 \ma 40oly2 = A - - k2) + 2gi a - k)- A = -~+M

Mgg= ; k = (gi/A)"3 K = 3g1k.m + MThe general solution shows that the motion is non-symmetrical bout theequilibrium position.To illustrate by a numerical example, if we take a=10ft., m=0.1lb.,M = 2.9 lb., a6o= 100ft./sec. nd g= 32.16 ft./sec2., henA = 33333 ft. /sec. 2 g = 0.0005465gi = 31.088 ft. /sec.2 = 0.000455k = 0.097702 ft. I c = 0.239 ft.

9.1123 8 sec.-2 X= 0.0234y2 = 0.5217 ft.2/sec2Then the period is P = { 1 + 6)/K} 1/22r = 2.08 sec

Solution by interpolation shows that the motion below the equilibriumposition requires 1.02 sec. per oscillation and that above 1.06 sec. The totalrange of oscillation is 0.478 ft., of which 0.235 ft. is below the equilibriumpositionand 0.243 ft. s above.SUGGESTED READINGS IN CONNECTION WITH"SUCCESSIVE GENERALIZATIONS IN THE THEORY OF NUMBERS"'

By E. T. BELL, Californianstituteof TechnologyR. Dedekind. Sur la theorie es nombres ntiers lgebriques,Bulletin desSciencesMathematiques Darboux'Bulletin), 2), vol. 1, pp. 17-41, 69-92,114-164, 07-248. Alsopublished eparately y Gauthier illars, aris, 1877.- Vorlesungenbber ahientheorie, on P. G. Lejeune Dirichlet, 4thedition, 894 (reprintedn 1912),Supplement I.G. Zolotareff.Theoriedesnombresntiers omplexes, vec une applicationaucalcul ntegral. t. Petersbourg, 873-4. Chapter II for heory f deals.Journales MathematiquesLiouville'sJournal),3),vol.6, (1880)pp. 51-84, 115-129,145-167.

1 This paperby Professor. T. Bell waspublishedn theFebruary umber f thisMonthly,vol. 34 (1927), p. 55-75.

campbell-a periodic solution for a certain problem in mechanics

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