ABSTRACT. m 11 m 10. A 1 A 3 A A4 C 6 L 2 C5 L 5 C 4 L 4 m 1 m 2 m 3 m 4 m 5 m 6 m 7 q 3 q 1 q 2 C 2 L 2 C 3 L 3

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1 UDC: STRUČNI RAD UTICAJ KRAJNJEG USPORENJA NAIZMENIČNE DVOUŽETNE ŽIČARE NA NAPREZANJE VUČNOG UŽETA EFFECTS OF BICABLE REVERSIBLE ROPEWAY ENDING DECELERATION ON DYNAMICAL STRAIN OF THE HAULAGE ROPE Dr Svetomir Simonović Visoka tehnička škola Bulevar Zorana Đinđića 5a, Novi Beograd REZIME Vučni sistem naizmenične dvoužetne žičare je modeliran kao materijalni sistem koji se sastoji od konačnih elemenata u obliku koncentrisanih masa i lakih opruga. Konstelacija konačnih elemenata i odgovarajućih generalisanih koordinata odražava uzdužne oscilacije masa izazvane krajnjim usporenjem. Primenjena je teorija malih oscilacija. Krajnje usporenje je simulirano postavljanjem početnih vrednosti generalisanih koordinata tako da zatezne sile u vučnim užadima uz pogonsku koturaču daju zahtevano usporenje kabina. Ova simulacija je učinjena pod pretpostavkom da se elastične deormacije vučnog užeta šire trenutno u odnosu na brzinu kabina. Na taj način su određeni zakoni kretanja materijalnog sistema što je sa svoje strane omogućilo da se odrede koeicijenti sigurnosti vučnih užadi na zatezanje. Koeicijent sigurnosti na gornjem kraju vučnog užeta je,66 a koeicijent sigurnosti na gornjem kraju zateznog vučnog užeta je 5,79 Ključne reči: širenje deormacija. ABSTRACT Haulage system o Bicable Reversible Ropeway has been modeled as a material system consisted o inite elements taking the orm o concentrated masses and light springs. The arrangement o inite elements and proper generalized coordinates relects longitudal oscillations o masses eectuated by ending deceleration. Theory o small oscillations is applied. The ending deceleration is simulated by setting the initial values o generalized coordinates so that tensile orces at the spans o traction ropes near the traction sheave produce reuested cabins deceleration. Aorementioned simulation is under supposition that elastic deormations o hauling rope spread instantly relative to cabins speed. That way law o motion o the material system is determined which in turn enabled determination o tensile saety actors. Tensile saety actor at the upper end o haulage rope is,66 and tensile saety actor at the upper end o tensile haulage rope is 5,79. Key words: deormations spread. m 5 m 4 6 m m 5 m 9 A A A A4 C 6 L 6 C5 L 5 C 4 L 4 m m m m 4 m 5 m 6 m 7 y C L C L x C L m 8 4 Figure. Model o Bicable Reversible Ropeway TEHNIČKA DIJAGNOSTIKA (BROJ 9) 7

2 . INTRODUCTION The haulage system o Bicable Reversible Ropeway is prone to oscillations due to combination o high elasticity o haulage ropes, weight o haulage ropes, weight o cabins and weight o counterweight that are exposed to change o speed during its starting acceleration. Under the term o haulage system o Bicable Reversible Ropeway are meant doubled haulage rope, cabins, doubled haulage tensile rope and counterweigh. The oscillations caused this way are deemed small in relation to overall dimensions o the ropeway, which enables linearization o dierential euations o the ropeway motion. Thereore, in this work, Matrix Theory o Small Oscillations is applied to explore the eect o ending deceleration on Dynamical strain o Haulage Rope.. MODEL OF THE BICABLE REWERSIBLE ROPEWAY HAULAGE SYSTEM In order to be determined strain o haulage rope because o oscillations caused by ending deceleration, particular case o Bicable Reversible Ropeway is modeled as material system with six degrees o reedom o movements as is shown at Fig.. The system is consisted o light springs and concentrated masses the arrangement o which and its generalized coordinates take into consideration longitudal nature o the oscillation. Here are the meanings o symbols used: m i i,...5 concentrated masses i i,...6 generalized coordinates A i i,,,4 suports X Ai, Y Ai i,,,4 cartesian coordinates o supports C i i,...6 stinesses L i i,...6 lenghts o hauling rope between supports β line angle corresponding L and L 6 β line angle corresponding L and L 5 β line angle corresponding L and L support 8 - concentrated mass - car - Counterweight - spring element Hauling rope is o type: 4,О-ГЛ-В-О-Н-96() ГОСТ with weigh per unit o lenght v,7 N/m; cross section area A v,6 cm ; and breaking orce F kid 4 kn Hauling tensile rope is o type: 5,5-ГЛ-В-О-Н-568(6) ГОСТ with weigh per unit o lenght v, N/m; cross section area A z,44 cm ; and breaking orce F kid 4 kn Supports tops coordinates are: (X A, Y A ) (, ) (X A, Y A ) (, -44) (X A, Y A ) (5, -6) (X A4, Y A4 ) (775, -676) Thereore, stinesses are: C 897 N/m C 4 N/m C 8496 N/m C 4 75 N/m C N/m C N/m The working deormation o the spring element between mass m and mass m is: st where st is static deormation o the spring element. The working deormation o the spring element between mass m and mass m 4 is: st where st is static deormation o the spring element. The working deormation o the spring element between mass m 5 and mass m 6 is: st TEHNIČKA DIJAGNOSTIKA (BROJ 9)

3 where st is static deormation o the spring element. The working deormation o the spring element between mass m 9 and mass m is: 4 4st 4 5 where 4 st is static deormation o the spring element The working deormation o the spring element between mass m and mass m is: 5 5st 5 6 where 5 st is static deormation o the spring element The working deormation o the spring element between mass m and mass m 4 is: 6 6st 6 where 6 st is static deormation o the spring element. POTENTIAL ENERGY OF THE BICABLE REWERSIBLE ROPEWAY HAULAGE SYSTEM Potential energy at the span corresponding L : ( st ) mg β E p C sin Potential energy at the span corresponding L : E p C m4g sin ( ) β st mg sin Potential energy at the span corresponding L : E p C ( ) st ( m6 m7 ) g sin β m5g sin Potential energy o the counterweight: E p 4 m8g4 Potential energy at the span corresponding L 4 : ( 4 4st ) m9g( 4 ) mg5 Potential energy at the span corresponding L 5 : E p 5 C4 sin β β β ( 5 5st ) mg5 sin β mg6 β Potential energy at the span corresponding L 6 : E p 6 C5 sin ( 6 6st ) mg6 β Total potential energy is: 7 E p E pi i E p 7 C6 sin As all generalized orces are eual to zero at the position o stable balance the ollowing conditions are obtained: p C st mg sin β C st mg sin β p C st C st m4 g sin β m5g sin β p C st C4 4st ( m 6 m 7 m 9 ) g sin β p C4 4st m9 g sin β m8g 4 m p 5 p 6 C C 5 5 st 6 6st g sin β C C 4 4 st 5 5st m g sin β m g sin β so, the potential energy can be expressed as: 4 4 E p c jk jk j k where coeicients o Stinesses are: c C C c C 4 c C C c C 8496 c C C c 4 * C4 * TEHNIČKA DIJAGNOSTIKA (BROJ 9) 9

4 c 5 C4 75 c 44 4 C c 45 C c 55 C4 C c 56 C5 448 c 66 C5 C The remainders are zeros. Masses are: m m L v 6,7,7 99,8 kg m m 4 L v 56,,7 66 kg m 5 m 6 L v 84,8,7 589,5 kg m 7 4 kg m 8 46 kg m 9 m L 4 v 84,8, 66,75 kg m m L 5 v 56,4, 7,5 kg m m 4 L 6 v 64,, 468 kg 4. KINETIC ENERGY OF THE BICABLE REWERSIBLE ROPEWAY HAULAGE SYSTEM E k Kinetic energy o the modeled haulage system is: [( m m ) & ( m m ) & ( m m ) & m & ] 4 [ m 9( & 4 & ) ( m m) & 5 ( m m ) & 6 ] Taking into consideration notation s s Ek a jk & j& k j k Coeicients o inertia are: a m m 99, ,8 a m4 m ,5 755,5 a m6 m7 m9 589,5 4 66,75 55,5 a 4 m 9 66,75,5 a 44 m 8 4m ,75 74 a 55 m m 66,75 7,5 967,9 a 66 m m 7,5 468,5 775, DIFERENTIAL EQUATION OF MOTIONS OF THE MODELED SYSTEM By using Lagrangian Euation o motion d dt k E E k p & r r r r,,, 6 linearized dierential euations o motions are obtained in matrix orm: [ a ]{ & } [ c]{ } {} where [a] [a ij ] i,j,, 6 and [c] [c ij] ] i,j,, 6 The linearized dierential euations o motions can be stated as { } [ A]{ & } {} where [ A] [ c] [ a] I solutions o the dierential euations o motions are suposed as ( ω α ) r µ r cos t r,,, 6 and i they are replaced in above stated system o dierential euation, the system o liner euation that is arived at is: ( Λ I [ A] ){ µ } { } which is euivalent to iteration suitable ormula [ A ]{ µ } Λ { µ } jr r,,, 6 r jr where Λ r is the reciprocal valueo the suare ωr o the angular speed o the r-th main oscillation µ - mode to which coresponds the modal column { } jr the r-th main mode o the material system oscillations. The iteration process is started by supposing arbitrary { } j µ and replacing it in iteration ormula to get new { µ j } and new Λ. ω The process is repeated until values o { µ j } and Λ are stabilized. The stabilized values are adopted as true values. 4 TEHNIČKA DIJAGNOSTIKA (BROJ 9)

5 6. ELIMINATION OF THE MAIN MODES OF OSCILLATION AND CORRESPONDING FREQUENCIES Let it be supposed that Λ > Λ >. > Λ s In order irst lower i be determined, current Λ and coresponding { ji } Λ i and { } µ to µ must be ji eliminated. Then, described iteration process tends to irst lower values o Λ and { µ }. It is well known rom small oscillation matrix theory that {} [ µ ]{ ξ} where: [ µ ] - main modes matrix { ξ } - column o he main generalized coordinates. It can be shown that [ A] m [ A] m [ S] m Now, the iterations are executed according to euation [ A ] { µ } { µ } m jm Λ m jm The iteration process is repeated until values o { µ jm } and Λ m are stabilized. The stabilized values are adopted as true values and Gaussian elimination method is applied in order m-th main mode o the oscillation to be eliminated. When m main modes o oscillation are determined the Gaussian elimination method yields:... s s... m m s s ξ j C j cos(ω j t - α j ) (j,,, s), s s so {} ξ []{} where [] [ µ ] Further, it can be shown that ( ) const( )[ a] r µ r where r is index o r-th main modes o the oscillation. In order m-th main mode o the oscillation to be eliminated, it is necessary that corresponding main generalized coordinate to be eual zero: ξ... m m m ms s where s 6 is a number o degrees o movement reedom o the material system under consideration. ( m) m... ( m) ms s From the last euation o the the Gaussian system it is obtained : ( m) ( m) ( m) ms m m m ( ) ( m ) m ( m ) so, this is how matrix [S] m look like: s Now the system is converging to m-th mode o oscillation, that is to the mode o oscillation that has the bigest next Λ. ( m)... ( m) ( m) ms... ( m) The main modes o oscillation are eliminated one by one starting with oscillation mode that has the biggest Λ When m-th oscillation mode is eliminated, the S is ormed that statisies the euation: matrix [ ] m TEHNIČKA DIJAGNOSTIKA (BROJ 9) 4

6 Applying exposed algorithm, the ollowing results are obtained: Λ.966E, s Λ,966 ω µ.e µ.8756e µ.47e µ 4.E µ E µ 6.74E µ E µ E Λ 5.967E- 9, s Λ5,967E ω µ E µ 5.E µ E- µ 45 -.E µ 55.E µ E Λ.8876E 4, s Λ,8876 ω 988 µ.88449e µ.e µ.84e µ E- µ 5 -.8E µ 6 -.6E Λ.78766E-, s Λ,78766E ω 574 µ E µ E µ.e µ E µ 5.488E- µ E Λ E- 4, s Λ4,49895E ω µ E µ E µ E µ 44.E Λ E-, s Λ6,765576E ω 6 4 µ 6 -.7E µ 6.567E µ E µ E µ E µ 66.E 7. LAWS OF MOTION OF THE MODELED SYSTEM Once main modes o the oscillation and the main reuencies o the system has been obtained, laws o motions o the modeled system can be determined by using initial conditions o the motion. The laws o motions stated in matrix orm are as: { } [ µ ]{ Acos ωt} [ µ ]{ Bsinωt} Constants matrices { A } and { } way: where: { ( ) } { A } [ µ ] { ( ) } { B ω } [ µ ] { & ( ) } B are detrmined this is initial generalized coordinates column matrix, TEHNIČKA DIJAGNOSTIKA (BROJ 9)

7 and { () } & is initial generalized velocities column matrix Also, ollowing euations are in orce: A j C j cos B where j,,.,6 j C j sin The ending deceleration j,7 m/s is simulated by setting the initial values o generalized coordinates so that tensile orces at the spans o traction ropes near the traction sheave produce reuested cabins acceleration. Aorementioned simulation is under supposition that elastic deormations o hauling rope spread instantly relative to cabins starting speed. This way is obtained: α α j j ( m m ) ( ) ( ) i j 6 C C6 6,6,7,96m so, initial conditions are,96.96,96,96,96 { } { & } i and cosinus coeicients matrix is: [K] [µ] [A dij ] i where elements o diagonal matrix [A dij ij ] are A dij ij A j i i j A dij ij i i j Taking into consideration the laws o motion o the modeled system or the constelation o initial conditions can be stated as: {} [K] {cosωt} dynamical orces in critical cross sections o the haulage rope can be determined. 8. CONCLUSION The working deormation o the spring element at the upper end o haulage rope is st thereore, the dynamical deormation is din,4 cos 574 (,586t),6 cos(,98t ),9 cos(, t) ( 4,5t), cos(9,855t),cos(, 4t ),6 cos and maximal dynamical deormation is din max,7 The maximal tensile orce at the upper end o haulage rope and tensile haulage rope is F st max m8 g 9,8 66 so, maximal statical deormation at the upper end o haulage rope is F st maxst C,8 Taking into consideration the breaking orce at the upper end o haulage rope is kid kid C F,895 tensile saety actor at the upper end o haulage rope is TEHNIČKA DIJAGNOSTIKA (BROJ 9) 4

8 kid,895 K,66 st din,8,7 max The working deormation o the spring element at the upper end o tensile haulage rope is 4 4st 4 5 so, the dynamical deormation is 4 din 4 5,cos 574 (,586t), cos(,98t ),8 cos(, t) ( 4,5t), cos( 9,855t ),4 cos(, 4t ), cos Maximal statical deormation at the upper end o tensile haulage rope is st Fmax st 66 4,99 m C4 75 Taking into consideration the breaking orce at the upper end o tensile haulage rope is kid Fkid 4 4 5, m C 4 75 Tensile saety actor at the upper end o tensile haulage rope is and maximal dynamical deormation is din 4 max,57 m K kid 4 st din 4 4max 5, 5,79,99,57 REFERENCES [] Беркман, Б. М., Бовский, Н. Г, Куйбида, Г.Г,. Леонтьев, С. Ю.: Подвесые канатные дороги, Машиностроение, Москва, 984. [] Дукелъский A.: Подвесые канатные дороги и кабельны краны, Машиностроение, Москва, 966. [] European Standard EN 97, Saety reuirements or cableway installations designed to carry persons-terminology, AFNOR, 5. [4] Radosavljević Lj.: Male oscilacije materijalnog sistema sa konačnim brojem stepeni slobode, Treće izdanje, Mašinski akultet u Beogradu, 986. [5] Radosavljević Lj.: Teorija oscilacija, Drugo izdanje, Mašinski akultet u Beogradu,968. [6] Simonović, S.:Analiza uticajnih parametara na dinamičko ponašanje ski lita, magistarski rad, Mašinski akultet u Beogradu, 995. [7] Simonović, S.: Dinamičko ponašanje dvoužetne žičare sa kabinama, doktorski rad, Mašinski akultet u Beogradu,. [8] Timoshenko, S.: Vibrations Problems in Engineering, d ed, Van Nostrand Company, Inc., New York, 955. [9] Tošić, S., Simonović, S.: Eects o Hard Stopping o the Ski Lit on Dynamical Strain o the Pulling Rope, FME Transactions, Vol., No., pp.-4,. [] Ungar, E.: Mechanical Vibrations, Mechanical Design Handbook, section 5, Rothbart, H as Editor, McGraw-Hill, TEHNIČKA DIJAGNOSTIKA (BROJ 9)