Balancing of Stephenson s Mechanisms

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1 Intenatona Jouna of uent Engneeng and Technoogy E-ISSN 77 0, P-ISSN 37 0INPRESSO, Rghts Reseved vaabe at Reseach tce aancng of Stephenson s Mechanss P. Neheah epatent of Mechanca Engneeng, Lods Insttute of Engneeng and Technoogy, ayatsaga, ydeabad-0009, Inda ccepted 0 May 0, vaabe onne 8 May 0, Vo., No.3 (June 0 bstact Ths pape deas wth the pobe of shakng foce and shakng oent baancng of hgh speed pana echanss. The desgn equatons and technques fo the copete shakng foce and shakng oent baancng of two types of sx-ba nkages due to both nea and otay neta but wthout consdeng extena oads ae deveoped. Shakng foce s baanced by the ethod of edstbuton of ass and shakng oent by the geaed neta counteweght and panetay-gea-tan neta counte- weght. These geaed neta counte weghts ae vey usefu fo baancng of ut-ba nkages. The poposed technques poduce bette esuts than that of the pevous technques. Keywods: Shakng foce, shakng oent, dynac baancng, vbaton, and Stephenson s echanss.. Intoducton echanca syste wth unbaanced shakng foce and shakng oent tansts consdeabe vbaton to the fae and foundaton of the achne. ffeent appoaches and soutons (G.G.Lowen et a 983 and V.akean et a 00 devoted to ths pobe have been deveoped and docuented fo vaous pana echanss. The dynac baancng of a echans s geneay acheved by two steps. st step s the canceaton o educton of the shakng foce and the second the canceaton o educton of the shakng oent. Geneay the canceaton of the shakng foce tanstted to the fae can be acheved by addng counteweghts n ode to ake the tota cente of ass of ovng nks statonay by addtona stuctues (V.Vande Wjk et a 0 o by eastc coponents (G.cet a 003. The shakng oent was baanced by the foowng ethods ( aancng by counte-otatons (R.S.ekof 973, J.L.eede 00, and V.VandeWjk et a 0 ( aancng by addng fou-ba nkages (.M.Gossen et a 00, Q. Jang et a 00 ( aancng by addng an neta fy whee otatng wth a pescbed angua veocty (V.akean et a.008, V.Van e Wjk et a 00 The pesent pape s the extenson wok of authos (V.akean et a.999, Gaoeng 989, Gaoeng 990. The esuts obtaned ae bette than the pevous ethod esuts. The pape s oganzed as foows: secton deas wth oducton, secton pesents atcuaton dyad. Secton 3 deas wth syetc nk wth thee otatona pas. ynac baancng of Stephenson s echanss s gven n secton. oespondng autho P. Neheah s wokng as Pofesso OI: Nueca exapes and esuts ae dscussed n secton. oncusons ae gven n secton.. tcuaton dyad. opete shakng foce and shakng oent baancng of an atcuaton dyad n open kneatc chan of two bnay nks and one jo s caed a dyad.when two nks ae atcuated by a jo so that oveent s possbe that aangeent of nks s known as atcuaton dyad. g. opete shakng foce and shakng oent baancng of an atcuaton dyad The faa schee of copete shakng foce and shakng oent baancng of an atcuaton dyad [0]-[3] s shown n g.. counteweght s attached to nk whch pets the dspaceent of the cente of ass of nk to jo. Then, by eans of a counte 83 Intenatona Jouna of uent Engneeng and Technoogy, Vo., No.3 (June 0

2 P. Neheah aancng of Stephenson s Mechanss weght wth ass cw [g.] a copete baancng of shakng foce s acheved. copete shakng oent baance s obtaned by fou gea neta counte weghts 3-, one of the s of the panetay type and ounted on nk.. opete shakng foce and shakng oent baancng of an atcuaton dyad by gea neta counteweghts ounted on the base The schee used n the pesent wok [g.] s dstngushed fo the eae schee by the fact that gea 3 s ounted on the base and s nked kneatcay wth nk though nk. Whee, T and T ae the nubes of teeth of the coespondng geas. Thus, the tota ass of the syste w be educed by an aount cw (3 ee the copete shakng foce and shakng oent baancng of the atcuaton dyad wth the ass and neta of nk taken o account s consdeed. o ths pupose ntay, statcay epace ass of nk by two po asses the hnges and and c at the centes of S S ( Whee, s the ength of nk, S and S ae the dstances between the centes of jos and and the cente of ass S of nk, espectvey. fte such an aangeent of asses the oent of neta of nk w be equa to g. opete shakng foce and shakng oent baancng of an atcuaton dyad by gea neta counteweghts ounted on the base To pove the advantages of such a baancng, the appcaton of the new syste wth the ass of nk not taken o account s consdeed. In ths case (copaed to the usua ethod g., the ass of the counte weght of nk w be educed by an aount 3 O cw ( cw 3 s the ass of gea 3, O s the dstance between the centes of hnges O and, cw s the otaton adus of the cente of ass of the counte weght. It s obvous that the oent of neta of the nks s coespondngy educed. If the gea netas ae ade n the fo of heavy s n ode to obtan a age oent of neta, the oents of neta of the gea neta counte weghts ay be pesented as I (=3. I S I ( S S S I s the oent of neta of nk about the cente S of ass S of the nk. Thus a new dynac ode of the syste s obtaned, whee the nk s epesented by two po asses, and has a I oent of neta. S Ths fact aows fo an easy detenaton of the paaetes of the baancng eeents as foows: W ( ( S W s the ass of nk, s the dstance between the centes of the hnges and, S s the dstance of the cente of hnge fo the cente ass of S of nk, s the otaton adus of the cente of ass W of the counteweght wth espect to, W [( W O OS ] W and (7 onsequenty, the ass of gea w be educed by an aount T ( ( 3O cw cw T s the ass of nk, OS s the dstance of the jo cente O fo the cente of ass S of nk. (8 W 3 O W 3 8 Intenatona Jouna of uent Engneeng and Technoogy, Vo., No.3 (June 0

3 P. Neheah aancng of Stephenson s Mechanss O, W3 s the otaton adus of the cente of ass of the counteweght. Takng o account the ass of nk bngs about the coecton n Eq.(3 n ths case, W (9 s the vaue decdng the change n the dstbuton of the asses of the syste nks esutng fo the addton of nk. 3. syetc nk wth thee otatona pas nk wth thee nodes s caed tenay nk, whee nodes ae pos fo attachent to othe nks. In the eae eseach by Gao eng eatng to baancng of nkages wth a dynac substtuton of the asses of the nk by thee otatona pas shown n g.3 two epaceent pos and ae consdeed. Ths esuts n the need to ncease the ass of the counte weght. oweve, such a souton ay be avoded by consdeng the pobe of dynac substtuton of nk asses by thee po asses. Usuay the cente of ass of such an asyetc nk s ocated nsde a tange foed by these pos., and,, ae po asses,, and ae the odu of adus vectos of coespondng pos,, and ae angua postons of adus vectos; s the ass of nk, I S s the oent of neta of the nk about an axs though S (axa oent of neta of nk. o ths syste of equatons the asses ae obtaned ; ; (0,, and ae detenants of the thd ode obtaned fo the above syste of equatons.. ynac aancng of Stephenson s echanss. Stephenson s echans wth two fxed pos (Mechans wth hgh degee of copexty The Stephenson s echans wth two fxed pos shown n g. s obtaned when one of the bnay nks n the basc Stephenson s chan s fxed. Ths s a echans wth hgh degee of copexty as oe than one ad of path cuvatue of oton tansfe pos and ae not known. Ths echans s used n the evesng gea of stea engnes. The baanced Stephenson s echans wth two fxed pos s shown n fg... Shakng foce baancng of the echans Lnk s epaced by dynac substtuton of nk asses by thee po asses, E and ; The condtons fo dynac substtuton of asses ae the foowng: e e E Ee E E 0 I S g.3 ynac substtuton of the asses of the nk by thee otatona pas The condtons fo dynac substtuton of asses ae the foowng: e e e 0 I S Whee, g. Stephenson s echans wth two fxed pos Whee, 8 Intenatona Jouna of uent Engneeng and Technoogy, Vo., No.3 (June 0

4 P. Neheah aancng of Stephenson s Mechanss, and E E ae po asses;,, ae the odu of adus vectos of coespondng pos., and E ae the angua postons of adus vectos. s the ass of nk. s the oent of neta of nk about an axs I S though S (axa oent of neta of nk ; Whee ; (, and, E E E ae detenants of the thd ode obtaned fo the above syste of equatons. Lnk s dynacay epaced by two po asses E and P and attached a counteweght W. o nk to be dynacay epaced by two po asses the condton to be satsfed s k ES P S, Whee, k s the adus of gyaton of nk about ts cente of ass ES s abtay fxed and P S condton W E P ( ES OS P S ( ( ES ES E OE P S P S s obtaned fo the above W Whee W P S OS s the adus of otaton of counteweght W Now nk s dynacay epaced by two po asses, P and nked kneatcay wth gea neta counteweght 9 by nk and attached a counte weght W aganst nk. o nk to be dynacay epaced by two po asses the condton to be satsfed s k S P S k s the adus of gyaton of nk about ts cente of ass, s abtay fxed and S P S P s obtaned fo the above condton. P S S ( ( P S P S S S Lnk s statcay epaced by two po asses, G G GS G O S G G fte statc epaceent of asses the oent of neta of nk s equa to IS I S S GS Now, nk s dynacay epaced by thee po asses O,, O O e O e ` e O 0 I S whee, O,, ae the po asses,,, O ae the odu of adus vectos of the coespondng pos, O,, ae angua postons of adus vectos, s the ass of nk. I S s the ass oent of neta of nk about ts cente of ass O O ; ; ( O ;,,, ae detenants of the thd ode obtaned fo the above syste of equatons. s the changed oent of neta of nk I S The counteweght attached aganst ass Gs equa to GO G W 9 W W whee ( ( W W9 S 8 Intenatona Jouna of uent Engneeng and Technoogy, Vo., No.3 (June 0 W W O W (3 s the counteweght attached aganst po ( s the adus of otaton of W P S S counteweght W Now nk 3 s dynacay epaced by two po asses,. 3 P3 o nk 3 to be dynacay epaced by two po asses the condton to be satsfed s k3 S 3 P 3S3 k 3 s the adus of gyaton of nk 3 about ts cente of ass, S s abtay fxed and s 3 P 3 S 3 obtaned fo the above condton.

5 P. Neheah aancng of Stephenson s Mechanss Now gea s ounted on the base and nked kneatcay wth nk 3 and nk 3 s statcay epaced by two po asses, j 3 JS3 3 S3 J J J The shakng oent geneated by the echans s baanced by usng the gea neta counteweghts 7-. whee J O fte statc substtuton of asses the oent of neta of nk 3 I I S3 S3 3 S3 JS3 I S 3 s the oent of neta of nk 3 afte statc substtuton of asses. The counteweght attached aganst ass J s equa to ( J O J W J W3 W ( 3 S3 ( 3 W3 W J W3 O W ( Whee s the counteweght aganst po ass J W J s the adus of otaton of counteweght W3 P 3 S 3 S3 W3 W po s the counteweght ass attached aganst g. aanced Stephenson s echans wth two fxed pos. Stephenson s echans wth thee fxed pos Stephenson s echans wth thee fxed pos has sx nks out of the fou ae bnay and two ae tenay nks. In the Stephenson s echans two tenay nks ae connected by bnay nks. Ths Stephenson s echans wth thee fxed pos shown n g. s obtaned when one of the tenay nks s fxed. Stephenson s sx-ba nkage can be thought of as two fou-ba nkages connected n paae and shang two nks n coon. Ths echans s osty used n Stea engnes. The baanced Stephenson s echans wth thee fxed pos s shown n g.7... Shakng oent baancng of the echans The shakng oent geneated by nks and ae gven n eq. (.The shakng oent geneated by the echans s detened by the su M M, M M M ( ae the shakng oents of the otatng nks and espectvey wth the neta of the epaced po asses taken o account. g. Stephenson s echans wth thee fxed pos In the Stephenson s echans wth thee fxed pos shown n g.. M M ( 3 ( I S ( I S3 W3 I OS S ( EOE W O W W W W W W J O J O GOG Whee, I S s the ass oent of neta of nk about ts cente of ass I I ae the changed oents of neta of nks 3, S3 S and espectvey. ae the angua acceeatons of nks and, espectvey g. 7 aanced Stephenson s echans wth thee fxed pos Lnk s the fxed tenay nk. The othe tenay nk 3 s connected to fxed tenay nk by bnay nks, and. The nks 3 and ae not decty connected to the fae. The geaed neta counteweghts equed to baance the shakng oent geneated by nks 3 and ae ounted on the base of the echans by 87 Intenatona Jouna of uent Engneeng and Technoogy, Vo., No.3 (June 0

6 P. Neheah aancng of Stephenson s Mechanss kneatcay nkng the geaed neta counteweghts and the nks by nks of known ass and cente of ass... Shakng foce baancng of the echans The shakng foce of the echans s baanced by dynacay epacng the nks by po asses and ass counteweghts. ee nk 3 s dynacay epaced by thee po asses by usng the foowng condtons + [ ] [ ] Whee, (7 ae the odu of adus vectos of coespondng pos ae the angua postons of adus vectos s the ass of nk 3 s the ass oent of neta of nk 3 about ts cente of ass and ae the thd ode detenants obtaned fo the syste of equatons. o nk to be dynacay epaced by two po asses the condton to be satsfed s ounteweght ( aganst nk can be obtaned as (9 Whee (, s the adus of otaton of counteweght o nk to be dynacay epaced by two po asses the condton to be satsfed s Whee s abtay taken and s obtaned fo the above condton fte nk s dynacay epaced by two po asses t s kneatcay connected to ts coespondng gea neta counte weght by nk, oe ove nk s statcay epaced by two po asses Whee s abtay taken and s obtaned fo the above condton ounteweght usng the foua aganst nk s cacuated by ounteweght aganst nk can be obtaned as ( (0 ( (8 Whee counteweght s adus of otaton of Whee (, s the adus of otaton of counteweght o nk to be dynacay epaced by two po asses and the condton to be satsfed s o nk to be dynacay epaced by the po asses and the condton to be satsfed s Whee s abtay taken and s obtaned fo the above condton Whee s abtay taken and s obtaned fo above condton 88 Intenatona Jouna of uent Engneeng and Technoogy, Vo., No.3 (June 0

7 P. Neheah aancng of Stephenson s Mechanss. Nueca exape ounteweght aganst nk can be cacuated as Whee counteweght ( (, s the adus of otaton of.. Shakng oent baancng of the echans The shakng oents geneated by nks,, and ae shown n eq. (.The shakng oent geneated by the nkage s detened by the su Whee ( ( ae the shakng oents geneated by nks,, and espectvey ae the ass oents of netas of nks, and espectvey s the changed oent of neta of nks espectvey. ae the angua acceeatons of nks,, and espectvey. o shakng oent baancng 8 gea neta counteweghts ae used, two at, two at and fou at. Shakng foce of the echans by the poposed ethod Poposed ( G 3 G3 G G G Shakng oent of the echans by the poposed ethod M poposed M M M M Shakng foce of the echans by Gaoeng s ethod ( Gaofeng ' ' G G 3 G3 G G G G G Shakng oent of the echans by Gaoeng s ethod M Gaofeng M M M M ( I S G E The Stephenson s echans wth thee fxed pos shown n g. has the foowng paaetes kg, k 0.3,.3kg, k 0.97, 3kg, k 0.73, 3.kg, k 0.89, kg, k 0.7, 0 9, GE., S, O 3, 3,., S 3,., 0ad / s, 0ad / s., E, 0 0.7, 0, 8, S., E 9, S.,. opason between the esuts of Poposed and Gaoeng s ethods The esuts of shakng foce n the echans by Poposed and Gaoeng s ethod ae shown n tabe.the esuts fo tabe show that except at cank ange 0 and 0 degees the shakng foces n the echans by poposed ethod ae ess than that of by Gaoeng s ethod and thee s a tte poveent n shakng foce baancng. The esuts of shakng oent n the echans by poposed and Gaoeng s ethods ae shown n tabe. On copason of shakng oent vaues t can be obseved that except at 0 and 0 degees of the cank ange, shakng oent vaues by the poposed ethod ae ess than that of by Gaoeng s ethod and consdeabe poveent n shakng oent can be notced. When shakng foce and shakng oent vaues ae copaed fo Stephenson s echans wth thee fxed pos t can be obseved that poved esuts ae poduced by the poposed ethod than that of by Gaoeng s ethod. Tabe Shakng foce copason of Stephenson s echans wth thee fxed pos ank ange (deg Shakng foce geneated n poposed ethod N Shakng foce geneated n Gaoeng ethod N Tabe Shakng oents copason of Stephenson s echans wth thee fxed pos ank ange(deg Shakng oent geneated n poposed ethod N- Shakng oent geneated n Gaoeng s ethod N Intenatona Jouna of uent Engneeng and Technoogy, Vo., No.3 (June 0

8 P. Neheah aancng of Stephenson s Mechanss The shakng foces and shakng oents of Stephenson s echans wth thee fxed pos ae detened at an eva of 0 0. t nube of postons of cank ange the esuts ae aost equa. Shakng foce s axu, N, at 0 0 and nu,.07 N, at 0 0. Shakng oent of the echans s axu, N-, at 0 0 and nu, N-, at 0 0. ut at hghe vaue of shakng oent.e. at 0 0, the poposed ethod offeed a geat poveent, 3% ove Gaoeng ethod. oncusons Shakng foce s baanced by the ethod of edstbuton of ass and shakng oent by geaed neta counteweghts. the panetay geas used fo baancng the shakng oent geneated by nks not decty to the fae ae ounted on the base of the echans, whch s constuctvey oe effcent and akes the baanced echans copact. Refeences G.G.Lowen,.R.Jepe, R.S.okof, 983, aancng of nkages an update, echanss and achne theoy, vo.8 (3, PP V.akean, M.R.Sth, 00, shakng foce and shakng oent baancng of echanss: hstoca evew wth new exapes, tansactons of the SME, Jouna of Mechanca esgn vo.7, PP V.Van de wjk, J.L.ede, 0,, synthess ethod fo nkages wth cente of ass at nvaant nk popantogaph based echanss, echans and achne theoy, vo.8,pp.-8. G.c, R.Shnzadeh sept-9 003, optu foce baancng wth ass dstbuton and a snge eastc eeent fo a fve ba paae anpuato, poceedngs of IEEE Intenatona confeence on Robotcs and utoaton, Tape, Tawan, PP R.S.akof 973, copet foce and oent baancng of nne fou-bas nkages, echans and achne theoy vo.8(3, PP J.L.ede,.M.Gossen 00, counte otau counteweght fo ght-weght dynac baancng. Poceedngs of SME 00 ET/IE confeence, sept.8-oct., sat ake cty, Utah, US PP.9-7. V.Van de wjk,. eeuenaee,.gossen, V.J.L.ede 0, copaatve anayss fo ow-ass and ow-neta dynac baancng of echanss, tansactons of the SME, Jouna of Mechanss and Robotcs vo., PP.-8..M.Gossen, G.cote, Y.Wu, 00, syntss and esghn of eacton ess thee d.o.f paae echanss IEEE tansactons on Robotcs & utoatons vo.0(, p.p Q.Jand, G.M.Gossen, 0,ynac optzaton of eacton ess fou-ba nkages, Tansactons of the SME, Jouna of dynac systes, easueent and conto vo.3, p.p. -. V.akenan, M.R.Sth, 008, esghnpf pana 3 O-RRR eactoness paae anpuatus, echatoncs vo.8, p.p 0-0. Van de wjk, J.L ede, ug -8 00, actve dynac baancng unt fo contoed shakng foce and shakng oent baancng, poceedngs of the SME 00 Intenatona n Engneeng confeence (IET/IE 00, Montea, Qubec, anada. Gaoeng, 989, opete shakng foce and shakng oent baancng of fou-types of sx-ba nkages, Intenatona Jouna of Mechans and Machne Theoy, Vo. (, pp akean V and Sth M R, 999. opete shakng foce and shakng oent baancng of pana nkages, Intenatona Jouna of Mechans and Machne Theoy, Vo.3, pp. -3 Gaoeng, 990, opete shakng foce and shakng oent baancng of types of fou-, fve-, sx-ba nkages wth psatc pas, Intenatona Jouna of Mechans and Machne theoy, vo.(, pp.83-9 Gaoeng, 99, opete shakng foce and shakng oent baancng of 7 types of eght-ba nkages ony wth evoute pas, Intenatona Jouna of Mechans and Machne theoy, vo.(, pp Intenatona Jouna of uent Engneeng and Technoogy, Vo., No.3 (June 0

Physics 1501 Lecture 19

Physics 1501 Lecture 19 Physcs 1501 ectue 19 Physcs 1501: ectue 19 Today s Agenda Announceents HW#7: due Oct. 1 Mdte 1: aveage 45 % Topcs otatonal Kneatcs otatonal Enegy Moents of Ineta Physcs 1501: ectue 19, Pg 1 Suay (wth copason