SOLUTION OF EQUATIONS OF MOTION FOR THE START OF AN ELECTRIC HOIST
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- Μακάριος Αναγνωστάκης
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1 Nmbe III, Vome VI, Jy OLUION O QUION O MOION OR H R O N LRI HOI Pae Vaní mmay: his conibion descibes a soion of eqaions of moion fo he sa of an eecic hois. o he cacaion hee ae sed aes fo he aca eecic hois, heeby i is ossibe o eify an accacy of he cacaion and sysem sensiiiy as we. In he second haf of his ae, hee ae saed he fis ess fo he sa of an eecic hois wih hee deees of feedom. Key wods: eqaion of moion, soion, eecic hois INROUION In fome isse of his jona, conceey in (, hee wee obained he eqaions of moion fo an eecic hois wih hee deees of feedom (O descibin is sa. his obained sysem of hee non-inea, second-ode diffeenia eqaions has no been soed in fome isse de o is comexiy. hs, aim of his ae is js he soion of eqaions in qesion.. OLUION O QUION O MOION. Recaiaion of he obem o he oses of bee oienaion in his obem is in he nex ae saed i., aen oe fom (. On he basis of i., hee ae obained menioned eqaions of moion ha wi be soed in he nex sbchae. esinaion of aiabes as we as hei o is idenica as in (. o, conside he eecic hois (see fi. wih mass m [] on which he wie oe wih enh [m] is ssended. ssmin ha wie oe is absoey iid and inanibe. he end of wie oe is fied he oad wih mass m [] and momen of ineia I [.m ] eaie o he G of oad. ecic hois is oweed by eamoo wih sain oqe M [N.m], is momen of ineia incdin oain as edced o in shaf is I M [.m ]. Raio beween ea moo and aein whees is i [-]. ecic hois has fo aein whees and each whee has a momen of ineia I 3 [.m ] and adis [m]. isance fom oad G o he ssension oin on he fiin is [m]. In. Pae Vaní, VŠ-echnica Uniesiy of Osaa, acy of Mechanica nineein, Insie of anso, eamen of anso and Pocess qimen, 7. isoad 5/7, Osaa Poba, e.: , -mai: ae.ani@sb.cz Vaní: oion of qaions of Moion fo he a of an ecic Hois 7
2 Nmbe III, Vome VI, Jy i. chemaic of he eecic hois wih swinin oad oce: ho, ( oowin eqaions of he moion (afe modificaions fo his soion wee deied by means of Laanian mechanics on he basis of i.. qaion of moion fo coodinae x i I 3 x m I M m m M i m sin( qaion of moion fo coodinae φ x m sin m ( ( sin( sin sin( ( sin sin qaion of moion fo coodinae ( I m m x ( ( sin ( sin( sin( sin sin(. Imemenaion of simifyin inaiabes o he oses of simificaion of he cacaion i is necessay o deemine simifyin inaiabes. hese inaiabes ae fonded on he basis of eecic hois and oad oeies, i.e. is mass, momen of ineia, ec. hese consan ae saed in he nex ae in ab.. ( ( (3 Vaní: oion of qaions of Moion fo he a of an ecic Hois 7
3 Nmbe III, Vome VI, Jy ab. Poeies of an eecic hois and oad Mass of an eecic hois m 63 [] Momen of ineia of he aein whee I 3,7859 [.m ] aein whee adis,7 [m] Momen of ineia incdin oain a edced o in shaf I M,5 [.m ] ain oqe of eamoo M,5 [N.m] Gea aio i 9 [-] Lenh of he wie oe [m] Load mass m [] Momen of ineia of he oad I,687 [.m ] isance fom oad G o he ssension oin on he fiin,38 [m] cceeaion de o aiy 9,8 [m.s - ] oce: ho On he basis of ab. ae deemined hese simifyin inaiabes i I 3 m I M m ( m (5 m (6 M i (7 ( I m (8 m fe imemenaion of hese inaiabes ino eqaions (, ( and (3 we e sin ( sin( ( sin( sin sin( ( sin ( ( sin ( sin( sin( sin sin( x (9 x sin x.3 Reaanin of eqaions Inasmch as a eqaions of moion conain coodinae x, i is ossibe o exess x ( fom eqaion (9 and seqeniay sbsie his obained eqaion ino eqaions ( and (. xessin x sin ( sin( x ( nd afe sbsiin we e ( ( Vaní: oion of qaions of Moion fo he a of an ecic Hois 73
4 Nmbe III, Vome VI, Jy sin sin ( sin( ( sin( sin sin( ( sin sin ( sin( ( sin ( sin( sin( sin sin( ( (3 ( fe eaanin of eqaions (3 and ( we e sin sin sin ( sin sin( ( ( sin( sin ( sin ( sin sin( ( ( sin( sin ( sin ( ( ( ( sin( (5 (6 qaions (5 and (6 aboe we can simify by nex inaiabe, hs (7 (8 I is aaen ha he manide and size of his inaiabes is idenica. fe sbsiin inaiabe (7 ino eqaion (5 and inaiabe (8 ino eqaion (6 we e sin [ sin sin( ( ] [ sin( sin ( ] sin (9 Vaní: oion of qaions of Moion fo he a of an ecic Hois 7
5 Nmbe III, Vome VI, Jy [ sin sin( ( ] ( [ sin ( sin( ] sin( ( ( sin(. Maix eqaion We can eqaions (9 and ( exess in a maix noaion in ode o obainin a maix eqaion. [ sin sin( ( ] [ sin sin( ( ] ( sin [ sin( sin ( ] [ sin ( sin( ] sin( ( sin ( sin( his sysem wi be soed by miyin inese maix o a maix on he ef side. he inese maix has a fom ( [ sin sin( ( ] [ sin sin( ( ] ( [ sin sin( ( ].5 djsmen fo ame s e se If we desinae: ( ( [ ( ( ( ( ] sin sin (3 sin ( [ sin sin( ( ] [ sin ( sin( ] ( [ sin( sin ( ] [ sin sin( ( ] sin( ( ( ( ( ( (5 Vaní: oion of qaions of Moion fo he a of an ecic Hois 75
6 Nmbe III, Vome VI, Jy sin sin sin [ sin sin( ( ] [ sin ( sin( ] [ ( ( ( ( ] [ sin( sin ( ] sin( ( ( sin [ sin sin( ( ] ( sin( [ sin sin( ( ] sin (9 ( sin( hen we obain fom eqaion ( on he basis of ame s e a maix eqaion (3 (3 If we now desinae: (3 (3 hen is (33 (3 hs, we obain a sysem of fo eqaions (35a (35b (35c (35d whee,,,,,, ae he fncions of aiabes and I sands o eason ha obained eqaions ose a sysem of fo, fis-ode diffeenia eqaions. onday condiions ae ( ( ( (, (36 becase of sisand a he beinnin. (6 (7 (8 Vaní: oion of qaions of Moion fo he a of an ecic Hois 76
7 Nmbe III, Vome VI, Jy We can he sysem descibed in eqaions (35a (35d wih bonday condiions (36 soe nmeicay, fo insance, by means of wo-se mehod. If we se a ime eiod Δ and if we desinae: Δ, ( (, ( and finay (, (37 hen we hae (38 (39 (, Δ (, and (, Δ (, ( nd nex Δ (a Δ (b ( ( Δ (c Δ (d.6 oion of coodinae x o menion, eqaion ( has a fom ( sin ( sin( x his eqaion may be eaaned ino fom x sin ( sin( ( fe sbsiin fom eqaion (3 we e x sin sin( sin sin( ( ( ( (3 Vaní: oion of qaions of Moion fo he a of an ecic Hois 77
8 Nmbe III, Vome VI, Jy x sin ( ( sin( ( onday condiions ae scceedin ( x( x (5 nd if we desinae: x x, x x(, x x ( (6 hen by aicaion of wo-se mehod wi be obained hese esimaes x x (7 x, x x Δ (8 x x x Δ (9 x x x Δ (5 whee x sin sin( (5 ( ( ( n accacy of nmeica soion deends on ime inea manide. he smae is inea Δ, he moe accae is he soion. he eo of soion is ooiona o sqae of ime inea manide ( Δ. If he momen of eamoo is sfficieny sma, hen wi be sma he anes φ and as we. o ha eason we can wie sin, ( sin (, ( ( (5 (53 hen we can simify he eqaions (9 and ( ( (5 (55 ( In addiion, becase of sma manide of he anes φ and, i is ossibe o eiminae a eemens in which ae hese anes esen, incdin hei fis-ode deiaies. hen we e Vaní: oion of qaions of Moion fo he a of an ecic Hois 78
9 Nmbe III, Vome VI, Jy Vaní: oion of qaions of Moion fo he a of an ecic Hois 79 (56 (57 We hae obained a sysem of wo inea diffeenia eqaions wih consan coefficiens. If we iniiae a sbsiions (58 (59 We e (6 If we miy second eqaion (fom a maix noaion (6 by nmbe R and o his eqaion wih fis eqaion fom a maix noaion, hen we e ( ( (6 ( ( (6 If we desinae, as he oos of eqaion (63 (63 ±, (6 hen we e ( ( (65 If we desinae (66
10 Nmbe III, Vome VI, Jy Vaní: oion of qaions of Moion fo he a of an ecic Hois 8 γ γ (67 We hae sin sin λ λ γ λ λ γ (68 fe sbsiion fom eqaions (58 and (59 we e ( sin λ λ ( sin λ λ (69 nd becase of ( ( ( (, we hae ( (7 ( (7 nd finay (7 (73 ecase and, we hae ( ( max (7 max (75 om Vièe s fomas is accodin o eqaion (6: < (76 hs, if we desinae oos so ha, > <, (77 and ( f (78
11 Nmbe III, Vome VI, Jy Vaní: oion of qaions of Moion fo he a of an ecic Hois 8 hen is ( < f (79 ( > m I m I f (8 Necessaiy has o be: < < < (8 fe sbsiion ino eqaions (7 and (75 we e he maximm aes of anes. i M max (8 max (83 Now, if we wan o esimae x, hen we hae o in eqaion ( eiminae sma ems ( sin, ( sin and if we se (, ( wheeby we e x (8 nd afe sbsiion ino eqaions (7 and (73 we e x (85 hs ( ( x (86 y means of ineaion wi be obain x, so ha ( ( sin sin x (87 fe he nex ineaion wi be obained x, so ha
12 Nmbe III, Vome VI, Jy Vaní: oion of qaions of Moion fo he a of an ecic Hois 8 ( ( x (88 ecase of ( ( x x, we obain fom eqaion (88 soion of consans of ineaion, so ha and ( (89 nd afe sbsiion we e he fina fom of x ( ( ( x (9.7 Recaiaion of obained eqaions eow ae saed he soed eqaions of moion fo coodinaes x, φ and. hese eqaions ae aid on he assmion ha he defecions φ and ae sfficieny sma. ( ( ( x. GRPHI RPRNION O OIN RLION On he basis of obained eqaions we can cay o ahic eesenaion of coodinaes deendin on ime and so obain he fis ess. o his ose wi be sed he aes fom ab. ha beons o an aca eecic hois (see i... heoeica execaions Imaine a see ba wih oeies (m, I and accodin o ab. ssended on he cane hoo. When an eecic hois aes, he oad as behind and swees o o a eae heih which means inceasin oenia eney. I is aaen ha in exeme osiion, whee he oad acceeaion is maxima and oad eociy is conesey minima, he oad oeshoos he wie oe; i.e. ae of ane of he oad defecion wi be eae han ae of ane of he wie oe defecion. hen, as we as
13 Nmbe III, Vome VI, Jy a he beinnin, he anes comae hemsees. Inasmch as i is no ossibe o incde a inds of esisances, he ocess cicaes isef and ies ae aes. oce: ho i. 3 mode of he measemen assemby wih an eecic hois and ssended oad. Gahs eow we can see he fis ime eaion, i.e. ane φ and ane deendin on ime. i. 3 ime-chaaceisic of defecion anes oce: ho Vaní: oion of qaions of Moion fo he a of an ecic Hois 83
14 Nmbe III, Vome VI, Jy In he i. 3 we can obiosy see oo ae aes of defecion anes. I is de o esisances which hae no been consideed. In addiion, i is ey diffic o incde esisance de o wie oe aanemen, aicay is iidiy. ain oqe of eamoo, conceey is ae is no easy o deemine as we. o wan of aes fom aca measin, i s no ossibe o comae obained daa wih ea daa. In he een ha he sain oqe of eamoo is decima, hen ae he imechaaceisics moe eaisic han he as, see i.. oce: ho i. ime-chaaceisic of defecion anes a he decima sain oqe i. 5 cceeaion of he wie oe end deendin on ime oce: ho Vaní: oion of qaions of Moion fo he a of an ecic Hois 8
15 Nmbe III, Vome VI, Jy i. 6 cceeaion of he oad.g. deendin on ime oce: ho i. 7 ecic hois acceeaion deendin on ime oce: ho chaaceisics saed aboe hae been efomed by means of an aoximae soion in M xce enionmen (see i. 8. his oammed soion maes i ossibe o chane any consan saed in ab. and accodin o hese consans immediaey e-cons he ess. Vaes of menioned consan ae enein ino he yeow ces. Vaní: oion of qaions of Moion fo he a of an ecic Hois 85
16 Nmbe III, Vome VI, Jy i. 8 n aoximae soion in M xce enionmen oce: ho y means of his aoximae soion we can obsee he infence of aica consan. 3. ONLUION On he basis of descibed cacaion, he soion of eqaions of moion fo he sa of an eecic hois wih hee deees of feedom has been efomed. eqeniay, accodin o obained daa, ae saed ime chaaceisics of soed aiabes. hese chaaceisics ae he fis ess of menioned sysem a a and ie sef infomaion fo he nex eseach.. KNOWLGMN his ae has been soed by he ojec P/9 on V-echnica Uniesiy of Osaa, zech Rebic. RRN ( VRNÍK, P. eiaion of eqaions of moion fo he sa of an eecic hois by means of Laane eqaions of he second ind. Pene s onacs [onine]., Vome V, Nmbe IV,. 8-85, [ci. -8-]. osné z <h://enesconacs.ce.cz/_/vani.df>. IN 8-67X. Vaní: oion of qaions of Moion fo he a of an ecic Hois 86
Appendix. The solution begins with Eq. (2.15) from the text, which we repeat here for 1, (A.1)
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