Te Interntionl Journl of TRANSPORT & LOGISTICS Medzinárodný čopi DOPRAVA A LOGISTIKA ISSN 45-07X APPROXIMATION OF A CATENARY FORMED OUT OF A ROPE ACCORDING TO THE HEIGHTS OF ITS POINTS OF HOLD Milivoj Vulić Univerity of Ljubljn, Fulty of Nturl Siene nd Engineering, Aškerčev, 000 Ljubljn, Sloveni milivojvuli@guetrnei Abtrt: Wen involved in field meurement, we frequently employ kind of rope or meuring tpe ftened to two different point nd treted out ro ditne In order for meurement uing ti tool to be more urte, it i neery to knowledge nd emine te gging tion of rope or meuring tpe wen ued in ti wy, inted of ting trigt line, it ume te pe of kind of urve, tenry, te prmeter of wi re influened by mny ftor, one of wi i lwy te eigt differene or lk tereof between te point were te rope i eld In ti pper, we ume tt mot rope ued for ti purpoe re of uniform weigt ditribution, undmged, nd of no eltiity, well ued in n idel environment of uniform eternl impt upon meurement, nd im to emine te effet of different eigt of point of old of rope upon te tenry it urve form, well ttempt to define nd nyle ti tenry by pproimting it into polynomil Key word: tenry, prmeter, djutment, polynomil pproimtion, tndrd devition INTRODUCTION An Elementry Definition of Ctenry Wen we tlk bout field obervtion obtined wit te id of rope or meuring tpe, eld in two different point nd t prtiulr eigt, it i neery to knowledge tt te rope or meuring tpe will g t point were it i not firmly eld, nd terefore lo ume prtiulr pe tt will not be trigt line, wi would invribly be te idel in u itution Ti pe ieved troug te gging of rope or meuring tpe i referred to tenry in mtemti [] [], nd i influened primrily by te trin in te point tt old te rope or meuring tpe in ple, it weigt, it weigt ditribution ie denity of different egment of te rope, nd te ditne between te two point were te rope i eld firmly ie te ditne over wi te gging i permitted to ppen A ny of tee ftor my influene field obervtion, we ve put mu effort into nlying tem, well nlying te tenry formed by te rope in u wy to minimie it effet on te meurement we obtin troug te ue of rope nd meuring tpe on te field It i uully poible to ompletely dird t let te effet of different weigt ditribution witin te rope, tt i more of n nomly tn regulr nd ignifint ourrene in rope nd meuring tpe Tu we ve inted foued on nlying ow tenry formed out of gging rope i determined by te differene in te point were it i eld firmly, preimrily teir eigt
Milivoj Vulić - Approimtion of METHOD A Metod for te Determintion of Ctenry A te rope or meuring tpe form gging pe, wi i eentilly form of urve, we ve pproimted te gging pe of te tenry to te form of urve we re lredy fmilir wit from mtemtil teory nd found tt te tenry orrepond loely wit te kind of urve formed by polynomil oniting of 5 term, wit 5 oeffiient Upon loer inpetion, owever, we found tt tere were only two oeffiient in ti polynomil tt, wen emined, preented wit n dequte impt nd degree of ury to tully be ble to determine te pe of te tenry We nturlly kept only tee two oeffiient in our pproimtion nd dirded te remining oeffiient, wi d n entirely negligible impt on te pe of te tenry, nd etenive eperimentl obervtion ve onfirmed tt ti polynomil wit two remining oeffiient mte te pe of tenry Determintion of Ctenry wit Point of Hold t n Equl Heigt We deided to firt emine te mot bi gging pe we would be ble to ome ro in our field obervtion, wi i tt of tenry formed between two point of old tt re t te me eigt, or rter tt ve eigt differene of zero ( ) In u e, te mimum degree of gging ppen t etly te mid-point of te ditne between te point of old, t Figure A tenry formed wen point of old re t equl eigt An epreion for te pproprite ditne orretion in u e i derived wit te id of tti blne ondition for rope burdened only wit teir own weigt [] A u, te blne 0 ondition in te diretion of te -i i nd tell u tt te tenile trin in every point of te tenry i ontnt, nd te blne ondition in te diretion of te y-i i v v v q 0 nd tell u tt te nge in tenion in te diretion of te y-i i proportionl to te peifi weigt of te rope, ie meuring tpe, or v q [4] An element of te tenry i epreed imply tn nd te inlintion ngle of te tenry lo imply y y /, y / / v [5] [6]
Milivoj Vulić - Approimtion of Knowing ti, if we inert te epreion for n element of te tenry, blne ondition of te y-i, y y / v q, into te epreion detiling te onequene of te, we get differentil eqution of te following form: / q y / v () If we ten tke te derivtive of te eqution for te inlintion ngle of te tenry, tn y / v /, nd ubtitute it into te eqution for n element of te tenry, y y /, we get noter differentil eqution, ti time of te following form: y / q y / () Te olution to ti differentil eqution i ten: y K () Here, i te prmter of te tenry, defined troug te epreion, i te integrtion ontnt, nd i te peifi weigt of te rope or meuring tpe If te oordinte origin i pled into te verte of te tenry, te integrtion ontnt i equl to Beue te gretet degree of gging of te rope i diplyed t, te epreion for te urve of te tenry n be written down in te following form: f (4) Te yperboli oine [7] n ten be developed into power erie wit te elp of te following epreion:! 4 4! n n! (5) And if we tke note of te firt two term of te erie nd te prmeter of te tenry, we ten get n epreion for te lrget degree of gging of te rope or meuring tpe, wi i: f q 8 (6) And te lengt of te tenry i ten: d dy d d (7)
Milivoj Vulić - Approimtion of Te yperboli ine [8] n ten lo be developed into power erie wit te elp of te following epreion:! 5 5! n n! (8) And if we tke note of te firt two term of te erie, we ten get n epreion for te lengt of te tenry, wi i: 48 4 q 8 f (9) Te orretion due to te gging of te rope or meuring tpe i terefore defined wit te id of te following epreion: l Q q 4 p 4 p (0) nd Here, i te weigt of te rope or meuring tpe, i te oberved lengt of te tenry, i te fore of te tenion of te rope or meuring tpe Determintion of Ctenry wit Point of Hold t Different Heigt We n now proeed to emine te more omple gging pe tt we would be ble to ome ro in our field obervtion, wi i tt of tenry formed between two point of old tt re t different eigt, or rter tt ve eigt differene tt i not equl to zero ( ) In u e, te mimum degree of gging doe not ppen t etly te mid-point of te ditne between te point of old, nd te epreion we derived bove for tenry formed between two point of old tt re t n equl eigt doe not old true [9] Figure A tenry formed wen point of old re t different eigt
Milivoj Vulić - Approimtion of 4 Te Generl Mtemtil Definition of Ctenry In order to be ble to better emine te bove itution, we need to ve omple undertnding of te mtemtil definition of tenry It i urve wit te pe of yperboli oine nd i mtemtilly defined wit te id of te following epreion: y e e () Wen we ue ti epreion, we preume tt te rope or meuring tpe tt tke te pe of ti urve i omogenou nd untretble In ti epreion, we employ te prmeter to mrk te eigt differene in te tenry, ie te prmeter i ued to define te verte of te urve in te point [0] Te urve i ymmetril in reltion to te y-i nd lie bove te urve of te prbol, i undertood wit te id of te following epreion: y () Figure A depition of tenry nd prbol 5 Our Appro to te Generl Mtemtil Determintion of Ctenry Wen we re deling wit two point of old tt re t different eigt, te tenry formed between tem n be determined wit te id of n djutment of term ppering in mtemtil erie Te ftor tt influene te pe of te tenry mot re te oberved ditne of te tenry nd te reltive eigt differene of te point of old of te tenry, nd te ltter i eily
Milivoj Vulić - Approimtion of tken from RTK-GNSS obervtion, te ury of wi i bolute in omprion to te ury of te meurement nd n tu be negleted A tenry i defined yperboli oine, te pe of wi n be pproimted wit it development into onvergent erie [] Tt i ow te beginning of ll 5 term of te erie mke up n epreion deribing te pe of it ord, ie it digonl ditne,,,0,,,,,0,, 0, 0, 0,0 0, 0,,,,0,,,,,0,, () Here, i te oeffiient of prtiulr term of te erie, wit indee nd tt determine te equentil term of te erie, i te lengt of te r of te tenry, nd i omputer-determined reltive eigt differene, gined from RTK-GNSS obervtionl dt In Tble, te mrti of oeffiient,, i diplyed E meured vlue belonging eqution of orretion, mde up from term of te erie nu j i n u j i 55 A (4) RESULTS Here, te bi for te determintion of te vetor of free term i te digonl lengt [] determined from te RTK-GNSS obervtion Figure 4 A tenry formed wen point of old re t different eigt
Milivoj Vulić - Approimtion of Tb A tble depiting te ontent of te mrti of oeffiient Te obervtion re ued to obtin te following epreion: GPS Y X H (5) Te vetor of te free term, f, i ten: 8 f 5 A GPS 55 5 T ij 5 (6) Tb A tble depiting te determintion of te vetor of free term f 5
Milivoj Vulić - Approimtion of In Tble, te wite for turning unknown nd meurement on nd off i tken into ount, nd te wite re denoted wit yellow ue Te oeffiient for te term of te erie re lo epreed, nd denoted wit blue ue Te digonl lengt gined from te previoulymentioned RTK-GNSS obervtion, determined wit te id of te epreion GPS Y X H, wile lo tking wite into ount, re epreed in te form of vlue benet te ell In ontinution, wen te mtri of oeffiient, A, nd te vetor of free term, f 5 55 bot determined, te mtri of oeffiient of norml eqution, n norml eqution, 55, re lo bot determined [5] tuly: 55, re N, nd te vetor of free term of N 55 A T A 55 55 (7) n 5 A T f 55 5 (8) Tb A mtri of te oeffiient of N nd te vetor of te free term of norml eqution n 5 55
Milivoj Vulić - Approimtion of Ti djutment by prmetri vrition [] i ontinued wit te ddition of n uilliry N mtri, 55, wit te id of wi te proe of djutment i utomtied, te determinnt of N te mtri of oeffiient of norml eqution, 55, i equl to zero, nd tt i wy it i not poible to diretly invert it nd determine te mtri of oftor of te ougt-fter vlue Q N NΔ Ti uilliry mtri, 55 55 55 A, i formed out of te mtri of oeffiient, 5 5, nd tu, wen it term re poitive, null or negtive, n uilliry vetor wit vlue of, or, repetively, i formed Ti mtri i ten gined wit te id of te epreion written down in te following tble: Tb 4 Te uilliry mtri N 55 NΔ N N full Te uilliry mtri, 55, i ten ounted into mtri 55, tereby gining mtri 55, wi i ten inverted Te reult of ti inverion i ten mtri of oftor of te ougt-fter Q vlue of 5 5 Te vetor of growt of te ougt-fter vlue 5 i ten lo determined, nd tu: 5 Q n 55 5 (9)
Milivoj Vulić - Approimtion of Tb 5 A mtri of te oftor of Q 55 Tb 6 Te vetor of growt of te ougt-fter vlue of 5 Tu, te finl reult of te diret djutment i te determintion of te vetor of orretion of te meured vlue 5 v : v 5 A 55 5 f 5 (0)
Milivoj Vulić - Approimtion of Te tndrd devition of te meured vlue i tu: 0 q o dig Q () Tb 7 A tble of te ougt-fter vlue of q Ten, te inpproprite term re ll dirded, under te ondition tt eiter i /, were wt fit i tt nd or rd term of te erie, or i /, were wt doe not fit i tt t or nd term of te erie Here, te oeffiient growt of te ougt-fter vlue 5 i, j re te oppoite vlue of te vetor of Tb 8 A depition of te proe of dirding of te term of te erie
Milivoj Vulić - Approimtion of Te firt dirded term i or /, te quotioent i / tke on te mllet poible vlue, ie i mller tn Te term re dirded in ti wy until te quotient doe not ume te vlue of or ny more, wi ten trnlte to probbility vlue of pproimtely 9545% Tu, term of te erie re derded under ti ondition nd reult, wt i obtined i n epreion for te lultion of genuine digonl ditne [4] Ti ditne n be lulted wit te id of te following epreion:,, 0 () If we ten tke te oeffiient we worked wit into ount, we get te following epreion out of te bove: 0, 0,0004 986 () Wit ti obtined digonl ditne, we re ten ble to djut te pproimte oordinte [5] we ve been working wit nd tu prtilly elude te influene of te gging of te rope or meuring tpe into te pe of tenry Te differene between tee pproimte nd djuted oordinte i preented wit te id of te following figure Figure 5 Te differene between te pproimte nd djuted oordinte
Milivoj Vulić - Approimtion of 4 CONCLUSIONS In n ttempt to minimie it influene upon our field meurement, we ve determined n pproprite pproimtion for tenry urve formed out of gging rope or meuring tpe ftened to two point of old nd treted out over ditne, wit point of old being eiter t n equl nd t different eigt Now tt we re wre of ti fundmentl pproimtion, it would be ueful to ontinue wit our invetigtion, ttempting to perfet our inigt into it by ontntly olleting informtion during field obervtion, entering it into ommon dtbe, tking re to ontntly onfirm nd perfet te teory beind it, nd invetigting te penomenon of gging rope forming tenry under different kind of ondition A i evident, wt my ppen wit rope wen it i tully ued on te field oppoed to only emined in teory i tt it my be different from te one tt we re ued to primrily nlying, mening tt it my even be rked or unoiling, or tt ondition my not llow for u to determine ertin pet of wt i ppening wit te urve of te rope nd mke tem idel, u te trin wit wi te point of old grip t te rope In u e, wile one prmeter involved in te formtion of tenry out of gging rope my not be eily determinble, we my till be ble to evlute oter prmeter nd ue tem to elp u better determine te tenry formed nd ow to del wit it Referene [] BLISS G A: 95 Clulu of Vrition Open Court Publ pter nd 4 [] GOLDSTINE H H: 980 A Hitory of te Clulu of Vrition (Berlin: Springer) [] PAPIČ, I et l: Elektroenergetk tenik I Ljubljn FE nd FRI Publition, 007 p 76 [4] MARENO, A et l: Te Stbility of te Ctenry Spe for Hnging Cble of Unpeified Lengt Europen Journl of Pyi IOP Publiing, 008 p 97 08 [5] FALLIS, M C: Hnging Spe of Non-Uniform Cble Amerin Journl of Pyi JP Publiing, 997 p 65 [6] CHATTERJEE, N et l: Te Hnging Cble Problem for Prtil Applition Atlnti Eletroni Journl of Mtemti, Volume 4, Number AEJM Publiing 00