The Study on the Cycloids of Moving Loops

Transcription

1 Journal of Applid Mathmatics and Physics, 018, 6, ISSN Onlin: ISSN Print: Th Study on th Cycloids of Moving Loops Gnnady Tarabrin Dpartmnt of Applid Mathmatics, Volgograd Stat Tchnical Univrsity, Volgograd, Russia How to cit this papr: Tarabrin, G (018) Th Study on th Cycloids of Moving Loops Journal of Applid Mathmatics and Physics, 6, Rcivd: March 5, 018 Accptd: April 1, 018 Publishd: April 4, 018 Copyright 018 by author and Scintific Rsarch Publishing Inc This work is licnsd undr th Crativ Commons Attribution Intrnational Licns (CC BY 40) Opn Accss Abstract Th infinit st of cycloids is cratd Each cycloid of this st is dfind as a movmnt trajctory of a point whn this point circulats on th conv closd contour of arbitrary form whn this contour movs rctilinarly without rotation on th plan with a vlocity qual to th tangntial vlocity of a point on circulation contour Th classical cycloid is lmnts of this st Th diffrntial quation of a cycloid st is drivd and its solution in quadraturs is rcivd Th invrs problm whn for th givn cycloid it is ncssary to fin th form of a circulation contour is solvd Th problm of diffrntial quation of th scond ordr with boundary conditions about a bnd of big curvatur of an lastic rod of infinit lngth is solvd in quadraturs Gomtry of th loop which is formd at such bnd is invstigatd It is discovrd that at movmnt of an lastic loop on a rod whn th form and th siz of a loop don t chang, ach point of a loop movs on a trajctory which namd by us th cycloid and which rprsnts a circumfrnc arch Kywords Curvs, Loops, Cycloids 1 Introduction Th work statd blow is not gnralization by somon bfor th cutd rsarchs and not dvlopmnt by somon bfor th offrd mthods It is compltly original work Togthr with it, this work isn t byond ral activity of th prson and consquntly it bass on all fundamntal knowldg savd up by popl Th contnt of this work was publishd in Russian [1] [] [3] This papr contains two indpndnt, but connctd with ach othr, parts Ths parts ar unitd that both thy can b classifid as th problms of diffrntial gomtry about infinit curvs with a loop which is formd by slf-crossing of branchs of ths curvs Diffrnc of parts is causd by th loops invstigatd blow ar mad of matrials of ssntially diffrnt proprtis DOI: 10436/jamp Apr 4, Journal of Applid Mathmatics and Physics

2 In th first part, th loop is formd by a flibl string Cross sction of such string hasn t th ffort namd bnding momnt Idal variant such string is a continuous chain formd from rings In th scond part, th loop is formd by idally lastic rod By ampl of such is a loop mad from a stl string of a guitar In cross sction of such rod at its bnd, thr is bnding momnt In both parts of this work, according to th author concption, invstigation of th loops form is not th main of articl sns Th loops form is studid in combination with movmnt of ths loops Such statmnt of a problm rsults in kinmatics rsarch of loops points Th loops kinmatics solvs problms about th form of movmnt trajctoris of loops points Th gomtrical sns of ths trajctoris is similar to sns of a classical cycloid, thrfor, in papr th points trajctoris of loops w calld cycloids Both parts of this papr invstigat gomtrical and kinmatic proprtis of abstract objcts Howvr, w will find out asily in thm an opportunity of practical applications, for ampl, for track of th transport machins, for tractors, for military quipmnt, for th problms solving of formation prvntion of hoss loops of divrs and astronauts and tc Cycloids of Circulation on Moving Contours Th mchanical modl which w will crat rprsnts mchanical systm which points mov on th cycloid studid by us As a rsult w will rciv mchanical systm which schm is rprsntd on Figur 1 For cration of modl w will prform th following oprations: tak intnsibl idally flibl string of infinitly big lngth, tak absolutly rigid rod which cross sction is limitd to th closd conv contour, wrap up string onc around a rod, and strtch this string by fforts P mad on infinity Thrfor, th string forms a flat loop that all its points without cption coincid with points of th dg of th rod cross sction Th intrsction point of th loop branchs lays on th contour, and at th sam tim this point lays on a straight lin along which th string is strtchd by fforts P This straight lin rprsnts th loop branchs with ndings rmot in infinity W will imagin that th contour (it is th dg of cross sction of absolutly rigid rod) movs without rotation on a straight lin coincidnt with strtchd string In th cours of th contour movmnt th string which wrap round this contour is sliding on th contour W will not by color tag a point on on of loop branchs in th intrsction point of loop branchs It will b that point which will bgin to mov on th contour Th color tag will mak a circulation (it will mak a full turnaround of contour) and stop movmnt in th intrsction point of th loop branchs Th color tag during of circulation will pass th distanc is qual to lngth of all loop At th sam tim th intrsction point of th loop branchs will go this distanc on th straight lin along which th string is strtchd Th dscrib movmnt of th color tag on th rod at forward movmnt of this rod rprsnts mchanical modl of circulation on a moving contour DOI: 10436/jamp Journal of Applid Mathmatics and Physics

3 Figur 1 Th schmatic viw of circulation on a moving contour W will plac mchanical modl in rctangular Cartsian coordinats, y so that th strtchd string coincidd with y-ais and that th loop thus is disposd on non-ngativ half of -ais (Figur 1) W will suppos that at th suprvision bginning momnt ovr contour movmnt th intrsction point of th loop branchs was locatd into th coordinat s origin O Not changing th form, th contour gos without rotation on a straight lin asid a positiv dirction of y-ais As th string all tim rmains strtchd th form of string loop will rmain constant, and th intrsction point of th string branchs at sliding a string on a contour during a contour movmnt will mov on y-ais W will discriminat th string branchs: th right-hand branch is curvd in th positiv sid of y-ais, its curvilinar part is disposd on a positiv half of y-ais and its rctilinar part is disposd on a ngativ half of y-ais; th lft-hand branch is curvd in th ngativ sid of y-ais, its curvilinar part is disposd on a ngativ half of y-ais and its rctilinar part is disposd on a positiv half of y-ais On Figur 1 w s: point O is branchs intrsction point, point A is join point of th right-hand and lft-hand branchs Point A is disposd on a positiv half of -ais Lt H b a distanc btwn points O and A W will dsignat diffrntly th quations of th right-hand and lft-hand curvilinar parts of loop Lt at th obsrvation bginning momnt of contour movmnt an quality y = p( ) b th quation of th right-hand loop branch and y = q( ) b th quation of th lft-hand loop branch Thn: p( ) > 0, q( ) < 0 whn 0 < < H and p( 0) = p( H) = q( 0) = q( H) = 0 Lt s mark by markr a point on a string on its rctilinar sit of th lft-hand branch W will dnot this point by lttr M Th contour gos to th right Thus th intrsction point of th string (loop) branchs coms narr to M-point At th obsrvation bginning momnt M-point coincids with th coordinat s origin O ( 0,0) At this tim momnt M-point bgins th movmnt on a contour countr-clockwis W will nam this movmnt th circulation Simultanously with it M-point will mak movmnt asid a positiv dirction of y-ais This movmnt is namd th transportation motion W will obsrv th M-point movmnt which will b suprposition of its both movmnts (circulation and transportation motion), and w will nam a trajctory of this movmnt (movmnts suprposition) th cycloid of circulation on a moving contour DOI: 10436/jamp Journal of Applid Mathmatics and Physics

4 Mathmatical mthods of this articl unit arn t byond traditional sctions of ttbooks of diffrntial and intgral calculus [4] Lt L b lngth of th contour lft-hand part from O-point up to A-point Th contour gos rctilinarly to th positiv sid of y-ais Whn th contour will pass th L-distanc, th M-point will pass by th contour lft-hand part from O-point up to A-point Taking into account both movmnt kinds (circulation and transportation) th markd M-point on coordinat s plan, y will b in A1 -point with coordinats = H, y = L Lt v b vlocity of transportation motion of M-point Movmnt vlocity of M-point on an arch lngth of th contour, it is tangntial vlocity, also will b qual to v This statmnt follows from th prsrvation law of a string quantity in an intrsction point of th string branchs: th string quantity which lft th right-hand part of a contour quals th string quantity which ntrd into th lft-hand part of a contour Thus it is ncssary to tak into account that th intrsction point of th string branchs gos with vlocity v Lt V = Vi+ Vyj b a vctor of rsulting vlocity of M-point which is qual to th vctor sum of tangntial and transportation vlocitis Lt α b an angl btwn a straight lin tangnt to th function graph y = q( ) and th -ais Componnts V, V y of a vlocity vctor V = Vi+ Vyj of M-point ar qual to 1 V ( ) 1 tan ( ), = vcos α, V = v + α 1 V 1 sin ( ) y = v + α Vy = v{ tan ( α) tan ( α) } According to gomtrical sns of th first drivativ function y q( ) hav tanα = q ( ) Taking into account it w driv 1 1 V = v( 1 + q ( ) ), Vy = v 1+ q ( ) ( 1+ q ( ) ) Lt y c( ) =, w = b th quation of a M-point movmnt trajctory and lt β b an angl btwn a tangnt to th diagram of this trajctory and -ais According to gomtrical sns of drivativ function y = c( ), w hav c ( ) = tan ( β ) According to physical sns of a vctor V = Vi+ Vyj, it is dirctd on a tangnt to a M-point movmnt trajctory Thrfor V Vy = tan ( β ) = c ( ) Substituting hr formulas V, V y w will driv th diffrntial quation of a M-point movmnt trajctory ( ) = ( ) + 1+ ( ) c q q (1) Intgrating this quation with th initial condition c ( 0) = 0, w driv th quation of th lft-hand half of th cycloid gnratd by th M-point movmnt on th lft-hand branch of a loop ( ) ( ) ( ξ) y = c q + 1+ q d ξ, 0 H () 0 In this quation th intgral with a variabl top limit dtrmins th currnt DOI: 10436/jamp Journal of Applid Mathmatics and Physics

5 valu of an arch lngth of th lft-hand half of a contour (th currnt valu of distanc which M-point passd on an arch of th lft-hand half of a loop) ( ) = + ( ) Th formula () can b writtn down in a form l 1 q ξ dξ (3) 0 y = q( ) + l( ) (4) This finit quation analytically dfins any of a cycloid of an infinit st th lmnts of which wr dfind abov by mchanical modl This rsult can b prsntd by th Thorm At movmnt of a circulation contour in a positiv dirction of an ordinats ais at a abscissa growth stag of th cycloid currnt point th ordinat of th cycloid currnt point is qual to th sum of two componnts: th currnt point ordinat of a circulation contour and th currnt arch lngth of th passabl contour part Symmtrical cycloids ar rsult of circulation on symmtric contours Th symmtry ais is a straight lin which is a prpndicular to a straight lin in which dirction th contour movs Th lft-hand and right-hand branchs of such contours submit to quality p( ) = q( ) Lt L b a lngth of th lft-hand loop branch Thn H 0 ( ) L= 1+ q ξ dξ Th Equation (4) dscribs a cycloid for y [ 0, L] symmtric cycloid for y [ L,L] y = L q( ) + l( ) Thn th quation of a is prssd by th formula (5) W will considr som spcial cass of a symmtric cycloid Circular (classical) cycloid (Figur ) is rsult of circulation on a circumfrnc Lt R b a circumfrnc radius Thn ( ) R + y = R is th quation of a circulation contour with H = R On formulas (), (3) w driv ( ) ( ) ( ) y = p = q R 1 R 1, 0 R, Calculations on formulas (4), (5) giv ( ) = π + arcsin ( 1) l R R Figur Circular (classical) cycloid-dashd lin, R = 1 DOI: 10436/jamp Journal of Applid Mathmatics and Physics

6 y [ 0, πr] y= R π + arcsin ( R 1) 1 ( R 1 ), 0 R ; y [ π R,πR] y= R 3π arcsin ( R 1) + 1 ( R 1 ), 0 R Traditionally classical cycloid is dfind by systm of th paramtrical quations In our cas th cycloid is unrolld along y-ais Thrfor in our cas th classical cycloid is dfind by th quations ( 1 cos ), ( sin ) = R t y = R t t Substitution of ths formulas in th first quation of a cycloid writtn down abov givs idntity Th idntity rsult mans that th dducd formula sts th lft-hand half of a classical cycloid Aftr that th quation of th right-hand half of cycloid dos not caus doubts Parabolic cycloid (Figur 3) is a rsult of a point circulation on a moving contour which is formd by th functions graphs ( ) ( ) ( )( ) y= p = q 4hH 1 H, 0 H, whr h-hight of a parabolic arch Currnt arch lngth such contour according to formula (3) is dfind on formula dξ 0 ( ) = + ( ) ( ξ ) l hh H On a formula (4) th diagram of a parabolic cycloid has bn constructd Elliptic cycloid (Figur 4) is a rsult of a point circulation on a moving contour in form of an llips Lt, ab b llips smi ais Thn ( ) a a + y b = 1 is th circulation contour quation with H = a and on formula () w driv Figur 3 Parabolic cycloid-dashd lin, H =, h = 15 Figur 4 Elliptic cycloid-dashd lin, a = 1, b = 05 DOI: 10436/jamp Journal of Applid Mathmatics and Physics

7 ( ) ( ) ( ) y = p = q b 1 a 1, 0 a Calculation on formula (3) rducs to an lliptical intgral { } ( ) ( ) ( ) ( ) 0 1 l = 1 1 ba ξ a 1 1 ξ a 1 d ξ, 0 a On a formula (4) th diagram of a lliptic cycloid has bn constructd Non-symmtrical cycloids ar gnratd by circulation on non-symmtrical contours Lack of a contour symmtry should b rlativ to a straight lin which is prpndicular to th dirction of th contour movmnt and mans that th inquality p( ) q( ) is carrid out For y [ 0, L] non-symmtrical cycloid quation is dscribd as in symmtrical cas by th formula (4) Th scond part of th non-symmtrical cycloid is dscribd by quation H 1 ( ξ) dξ ( ) 1 ( ξ) dξ y = + q + p + + p 0 Invrs problm about cycloid of circulation on a moving contour appars if in q and c( ) chang rols with ach othr Solving q q, w driv Equation (1) functions ( ) th Equation (1) with unknown ( ) and intgrating ( ) H 1 d q( ) = c( ) C + c ( ), (6) whr C is arbitrary constant This formula givs th rason to pos a qustion about th solution of an invrs problm about cycloid of circulation on a moving contour: to driv th quation of a moving contour from th givn a cycloid quation Th formula (6) dfins th quation of th lft-hand part of a circulation contour within arbitrary constant C-summand To find unknown (rquird to us) th contour quation, it is ncssary to calculat th valu of C-constant Th cycloid is a trajctory of th movmnt of th markd point which circulats on a contour and a contour at th sam tim movs rctilinarly without rotation So any bginning of movmnt of th markd point is a point of a cycloid origin and is also a point of circulation origin on a contour Consquntly in th invrs problm considrd hr th coordinats of a point of a cycloid origin should b givn togthr with th cycloid quation If in th quation (6) to mak substitution c( ) and to cut intgration w will driv a formula of th curvs st for th givn cycloid This st contains rquird curv If aftr that to rplac variabls, y by th point coordinats of a cycloid origin thn w will driv th quation th solution of which will giv us th C-constant valu corrsponding to th givn cycloid Thus th invrs problm will b solvd and this solution will b only on For ampl, lt th symmtric cycloid is givn by function ( ) ( ) a + b b a if 0 a, c( ) = a + b + b a b if a < a DOI: 10436/jamp Journal of Applid Mathmatics and Physics

8 Hr th cycloid quation is givn and simultanously with it th origin point ( 0,0 ) of th curv is givn Calculation by th formula (6) givs formula b a if 0 a, y = p( ) = q( ) b b a if a < a This formula dtrmins a contour in th form of a rhombus with smi-as a, b On Figur 5 th givn cycloid and a rhombus inducing it ar showd, whr a = 15, b= 1 3 Loop on Elastic Infinit Rod Statmnt of lastic loop problm Th lastic rod of infinit lngth is strtchd by forcs P which ar applid on indfinitly far th rod nds On th rod thr is an lastic loop This loop is formd by singl fold slf-crossing of two half of th considrd rod Th lastic loop was studid by L Eulr [5] But it was th loop which was formd at longitudinal comprssion of a rod of finit lngth This loop is namd Eulr s lastic W will plac th rod with th loop in rctangular Cartsian coordinats, y so that th rod will sttld down along y-ais and th loop was dirctd to th positiv sid of -ais Thus y-ais will b th curv asymptot rprsnting th rod with th loop Symmtry ais of th loop w will combin with y-ais W will discriminat two loop branchs: th right-hand branch which is curvd in th positiv sid of y-ais (its lft nd is strtchd in ngativ infinity), and th lft branch which is curvd in th ngativ sid of -ais (its right nd is strtchd in positiv infinity) On Figur 6 w s: O-coordinat s origin, A-branchs intrsction point of th loop, B-join point of th right-hand and lft-hand of th loop branchs Lt h b a distanc btwn O-point and A-point, w will nam h th intrsction point lvl; lt H b a distanc btwn O-point and B-point, w will nam H th loop hight Lt y = ( ) b an quation of right-hand loop branch Thn ( ) < 0 whn 0 < < h, ( ) > 0 whn h< < H, ( h) = ( H) = 0, lim ( ) = 0 W will assum that during formation of a loop th rod was bnt undr laws of dformation of an lastic thin bam Lt E b Young s modulus of a rod matrial Figur 5 Rhombic cycloid-dashd lin DOI: 10436/jamp Journal of Applid Mathmatics and Physics

9 Figur 6 Loop on lastic infinit rod Th rod which w study is a thin bam which bnd thory is cratd with us of th simplifying Brnoulli s hypothss [6]: 1) Hypothsis about flat sctions: a cross sction which is flat at th not dformd bam rmains flat during a bnd and maks thus rotat around of a nutral ais; ) Hypothsis about th linar strss condition: normal strss σ and rlativ linar strain ε of longitudinal fibrs of a rod ar intrlinkd by Hook s law σ = εe ; 3) Hypothsis about vn distribution: rod fibrs which ar quidistant from a nutral ais hav qual strsss W will dsignat a symbol I th inrtia aial momnt of a rod cross sction in a bnd plan Equation of th lastic loop form Curv curvatur in th currnt point of th bnt rod ais within th framwork of Brnoulli s hypothss is dirctly proportional to th bnding momnt which is acting in cross sction of a rod of this ais point For th right-hand branch this statmnt is prssd by quality { } 3 ( ) 1 ( ) + = P EI, (7) which is th diffrntial quation of th lastic loop form W will ntr a nw variabl z( ) = ( ) thrfor th quation (7) will b transformd to th quation of th first ordr ( ) 1 ( ) 3 z + z = P EI W will mak rplacmnt of a variabl z( ) sinh t( ) and w will intgrat Th common solution of this quation is { } 1 ( ) ( ) = 1+ = P EI + C, (8) whr C is arbitrary constant On Figur 6 w s, that on right-hand branch ( ) + whn 0+ 0 W subordinat th Equation (8) this condition and in rsult w driv C = 1 Th lastic rod with a loop which w amin is vrywhr th smooth figur symmtric rlativ to -ais Thrfor on right-hand branch ( ) DOI: 10436/jamp Journal of Applid Mathmatics and Physics

10 whn H 0 W subordinat th quation (8) with C = 1 this condition and in rsult w driv P = 4EI H (9) W will substitut C = 1 and P = 4EI H into th quation (8) and solv In rsult w will hav this quation about unknown ( ) 1 1 ( ) = ( H) 1 ( H) ( H) 1 ( H), 0< < H (10) W will mak substitution t 1 ( H) = It will allow us to hav 1 { } ( H) 1 ( H) d Hln 1 1 ( H) = + ( H ) W shall mak substitution ( ) t= 1 H It will allow us to hav 1 ( H) 1 ( H) d= H 1 ( H) Aftr that w will find uncrtain intgral from function (8) W will tak a constant of intgration such that condition ( H ) = 0 was satisfid It will b function which will rprsnt th finit quation of th right-hand part of an lastic rod ais with a loop ( ) H + y= ( ) H 1 ( H) ln, 0 < < H (11) H Obviously, th point ordinats of th lft-hand loop branch will hav opposit signs Gomtrical charactristics of lastic loop W will construct a loop-1 by hight H1 = H Th quation of this loop will b (11) Lt s tak on a loop-1 any way two points (, ), (, ) 1 M a b M a b At substitution of points coordinats M, M into Equation (11) w will driv two th idntitis which will hav of a form ( ) ( ) 1 b= H 1 ( ah) ln 1 1 ( ah) ah + (1) Lt s construct diffrnt loop it will b a loop- by hight H = kh whr k = const > 0 Th quation of this loop will b ( ) kh + y = ( ) kh 1 ( kh ) ln, 0 < < kh (13) kh W will tak two points (, ), (, ) N ka kb N ka kb At substitution of points coordinats N1, N into Equation (13) w will driv two th idntitis which will hav a form also (1) Hnc points N1, N lay on th loop- Now w will calculat distancs btwn points on both loops ( ) ( ) MM = a a + b b, DOI: 10436/jamp Journal of Applid Mathmatics and Physics

11 ( ) ( ) ( ) ( ) N N = ka ka + kb kb = k a a + b b As a rsult of comparison of ths distancs w driv NN 1 MM 1 = k= H H1 This quality mans, that th loop-1 and a loop- ar gomtrically similar figurs, thir similarity cofficint is qual to k = H H1 This fact givs th rason to formulat th following fundamntal charactristics of loops on lastic infinit rods Thorm All loops on lastic infinit rods ar in pairs th gomtrically similar figurs with th similarity cofficint qual to th ratio of thir hights Similarity of loops opns an opportunity to dduc formulas of gomtrical forms and ths formulas will b suitabl for rods of any sourc data valus P, E, I W hav by formula (9) H = EI P W will talk th unitary loop if H = 1 It can b at P = 4EI Th branchs intrsction point lis on -ais Thrfor ( h ) = 0 Substituting h= ηh into th formula (11), w driv 1 η = 1 ln 1+ 1 η η (14) ( ) ( ) W will solv this quation In rsult of it w will hav η 088 η-numbr is th lvl of th branchs intrsction point of th unitary loop W will nam η-numbr th unitary lvl Th currnt lngth of a loop arch with th origin in th intrsction point A is dtrmind by th formula ( ) ( ) l = 1+ d, h H h Substituting hr formula (10) and calculating quadraturs w driv ( ) ( ) l( ) = H π ( 1 ) ln 1 1 ( H) H +, h H, (15) whr π is som constant W will calculat it As th origin point of th currnt lngth of a loop arch is in th intrsction point A, thn l( h ) = 0 W will substitut h= ηh into th formula (15) and will us th formula (14) In rsult from th quation l( h ) = 0 w will driv π 1 ln 1+ 1 η η = 0 π 1 η = 0 ( ) ( ) π + η = 1 (16) As w alrady calculatd η-numbr, from this quation w dtrmin π 0958 DOI: 10436/jamp Journal of Applid Mathmatics and Physics

12 By formula (15) ( ) π π L= l H = H = LH Numbr L is qual to a half loop lngth Consquntly π -numbr is th quotint of a half loop lngth to hight of this loop If H = 1, thn π -numbr is half lngth of unitary loop π -numbr w will nam th π-numbr of an lastic loop Th loop branchs ar crossing with angl γ Making us th formula (10) at = ηh w driv ( H) γ = arctan η γ 068π 113 Numbrs ηπ,, γ ar absolutly constants of th lastic loops on infinity rod Th Equation (16) is th gomtrical idntity of an lastic loop s btwn a loop branchs whn h is dfind by Th currnt ara ( ) formula s = ξ dξ ( ) ( ) h 1 1 ( H) + s( ) = ηπ + 1 ln H, h H H H H From hr th ara of an lastic loop is qual to S = ηπ H Cycloid of lastic loop Th infinit lastic rod with loop is mchanical systm Th loop of this systm can mov forward without th form chang on a straight lin paralll to th loop asymptot (paralll to y-ais) In practic it can b obsrvd, for ampl, on air hoss for divrs Som of physicists ar inclind to bliv that loop movmnt on infinit lastic rod is a soliton (solitary wav) [7] W will considr that th loop gos to th positiv sid of y-ais On a rod on th positiv sid of y-ais w will mark M-point Whn th branchs intrsction point will com in M-point, th M-point will start movmnt on a loop This movmnt of th markd point on a loop will rpat all th movmnt laws which w considrd abov in sction Circulation cycloids on moving contours Thrfor now w hav th right to tak rsults of that sction and w mak us of that Into th cycloid Equation (4) w will mak th following two rplacmnts: th lft-hand half of an lastic loop y = q( ) ( ) in th form (11) and th currnt arch lngth l( ) of an lastic loop in th form (15) As a rsult w will driv th quation of th first half of th circulation cycloid on an lastic loop y= H π 1 ( ) H, h H (17) DOI: 10436/jamp Journal of Applid Mathmatics and Physics

13 Th considrd cycloid is symmtric Thrfor from th formula (5) w driv th quation of th cycloid scond half of an lastic loop y= H π 1 ( ) + H, h H (18) W will transform th Equations (17), (18) to th form ( ) ( π ) 0 + y H = H This quation is th canonical circumfrnc quation of radius H with th circumfrnc cntr into a point with coordinats = 0, y = π H Th cycloid origin is th point ( h,0), th cycloid nding is th point ( h,π H) In Figur 7 thr is th lastic loop cycloid (dottd graph) and thr loop positions (firm lins) which w hav mad by formulas (11), (17), (18) 4 Conclusions Gomtry of movmnt trajctory of a point at its circulation on rctilinarly moving contour is invstigatd W hav namd such trajctoris cycloids by analogy to a classical cycloid which is a point trajctory of a circumfrnc whn th circumfrnc rolls on a straight lin without sliding On kinmatics of mchanical modl of circulation on a moving contour, w hav formulatd th gnral dfinition of all sts of considrd cycloids W hav drivd th diffrntial quations of ths cycloids and hav solvd in quadraturs th boundary problms of th drivd quations Cycloids curvs for particular forms of circulation contours ar constructd W hav solvd th invrs cycloid problm of point circulation on a moving contour This is th problm about finding of a circulation contour for givn cycloid quation Rsults of rsarch of an lastic loop hav practical significanc as th loops on th lastic infinit rods cit spcial attntion in many tchnological procsss and tchnical dcisions It is stablishd that an lastic loop on an infinit rod posssss marvlous gomtrical proprtis To fl it, w compar an lastic loop to a circumfrnc All loops on th lastic infinit rods by pairs ar th gomtrically similar figurs with th similarity cofficints which ar qual to th quotint of thir hights H Figur 7 Cycloid of lastic loop DOI: 10436/jamp Journal of Applid Mathmatics and Physics

14 All circumfrncs by pairs ar th gomtrically similar figurs with th similarity cofficints which ar qual to th quotint of thir radii R Half lngth: of a loop quals to L= πr Ara: of a loop quals to S L= π H, of a circumfrnc quals to = ηπ H, of a circl quals to S = πr Arch of a circumfrnc is a cycloid of a loop on th lastic infinit rod Th rsult of comparison btwn a loop and circumfrnc givs th basis to nam a loop on an lastic infinit rod, th sibling of a circumfrnc Rfrncs [1] Tarabrin, GT (01) Loop on Elastic Rod Stroitlnaya Mchanika i Rascht Sooruzhniy, 3, (In Russian) [] Tarabrin, GT (01) Circulations Cycloids on Moving Countrs Stroitlnaya Mchanika i Rascht Sooruzhniy, 4, (In Russian) [3] Tarabrin, G (014) Non-Chrstomathy Problms of Mathmatical Physics Palmarium Acadmic Publishing, Saarbruckn (In Russian) [4] Piskunov, N (1975) Diffrntial and Intgral Calculus Vol 1, Mir Publishrs, Moscow [5] Rabotnov, JN (1988) Mchanics of Dformabl Solid Nauka, Moscow (In Russian) [6] Timoshnko, S (1955) Strngth of Matrials D van Nostrand Company, Inc, Princton, Nw Jrsy, Toronto, Nw York, London [7] Filippov, AT (1990) Multiform Soliton Nauka, Moscow (In Russian) DOI: 10436/jamp Journal of Applid Mathmatics and Physics