ANALYSIS OF VIBRO-IMPACT PROCESSES OF A SINGLE-MASS SYSTEM WITH VISCOUS DAMPING AND A SINGLE LIMITER

Transcription

1 Ljubiša Garić hs://doi.org/0.78/tof.4303 ISSN ISSN NLYSIS OF VIBRO-IMPT PROESSES OF SINGLE-MSS SYSTEM WITH VISOUS MPING N SINGLE LIMITER Summary Th ar rsns an anaysis of h horizona sraigh-in moion of a g-mass, on-sidd vibro-imac sysm in h cass whn h rna corciv forc and h viscous daming forc ar known and a riodic vibro-imac mod is raizd in h sysm. Vibroimac sysms rrsn h basis of indusria machinry. Priodic moion mods, raizd as forcd damd osciaions, ar characrisic of hs sysms. u o h variy of vibroimac sysms, hr ar various comuing mhods usd for h anaysis and dscriion of rocsss in vibro-imac sysms. Th rocss of obaining rsus is, gnray, vry com and im-consuming bcaus hr is a nd of ug inar diffrnia quaions in comuing. Th aim of h anaysis rsnd in his ar is o drmin h aras of h riodic vibro-imac mod isnc. Ky words: vibro-imac osciaor, vibro-imac mods, rna corciv forc, viscous daming forc.. Inroducion Th subjc of h rsarch conducd in his ar is h anaysis of h horizona sraigh-in moion of a g-mass, on-sidd vibro-imac sysm, whn a riodic vibroimac mod is raizd in h sysm. This anaysis is conducd ug mahmaica moding of vibro-imac syms. For his uros, i is assumd ha h rna corciv forc and h viscous daming forc ar known, whi i is ncssary o drmin condiions (aras of h riodic vibro-imac mod isnc. Two ams (cass ar ord: h firs, whn h rsiuion cofficin is R = 0.7, daming iss, and h daming dcrmn is 0 ; h scond, absouy asic imac, wih h rsiuion cofficin R = and wih no daming, 0. In boh cass, rsus nabing h sabiiy of vibro-imac sysm moion ar obaind. Th aim of h rformd rsarch is an incras (imrovmn in h riodic vibro-imac sysm funciona fficincy, basd on h mahmaica mod of h sysm. This ar considrs and rsns h rsus obaind for a dfind form of h vibroimac sysm srucur (wih a g-mass, a g imir and on sring, which is osd o h infunc of a riodic rna forc. Th obaind rsus can giv som idas and guidoss for rsarch ino vibro-imac sysms wih diffrn srucurs and diffrn ys of rna forc. TRNSTIONS OF FMEN XLI-3 (07 9

2 Lj. Garić naysis of Vibro-Imac Procsss of a Sing-Mass Sysm wih Viscous aming and a Sing Limir Th mos imoran scinific rsus which imrov h knowdg abou h dynamics of vibro-imac sysms hav bn ubishd in scinific iraur, mos noaby in [], [] and [3]. In hs ubicaions, h anaysis of vibro-imac sysms is rformd assuming ha sysm aramrs and h rna corciv forc ar known; subsquny, corrsonding mods ar dfind and condiions for h isnc of hs mods ar amind. In [], Babickii V.I. amind h isnc of h riodic vibro-imac sysm mod for h im inrva wih a dfind riod of osciaion. This was a guidos which was foowd in h rsarch rsnd in his ar. haracrisics of h horizona sraighin moion of g-mass, on-sidd vibro-imac sysms ar rsnd in [4], [5], [6], and [7]. vibro-imac sysm wih wo masss moving saray and imacing muuay is rsnd in [8], whi a vibro-imac sysm wih a movab imir is rsnd in [9]. Sabiiy considraions of h riodic moions of vibro-imac sysms ar givn in [0]. In [], vibro-imac sysms wih a g imir as w as wih wo imirs fid symmricay on boh sids of h osciaor ar rsnd and dscribd. Th am rsnd in [] is a sysmaic sudy on a riodicay forcd osciaion sysm wih imac. Vibro-imac sysms ar widy usd in mchanica nginring. Usuay, riodic vibro-imac sysms ar usd whn im inrvas bwn wo conscuiv imacs ar h sam. Undr his condiion, imacs ar rformd during h forcd sysm moion, which roducs various ffcs. ynamic hnomna occurring as h rsu of imac can roduc harmfu or bnficia ffcs. Th harmfu ffcs (damag nd o b rducd, whi h bnficia ffcs nd o b magnifid in ordr o imrov h rocss roduciviy.. Mahmaica mod of a g-mass on-sidd vibro-imac sysm In h am shown bow h dynamics of a g-mass, forciby damd on-sidd vibro-imac osciaor is sudid in h cas whn i is osd o h acion (infunc of h foowing facors: an asic forc F c, a viscous daming forc Fw b, and an rna riodic corciv forc F( F0. Thrby, h mass of h osciaor m, h sring siffnss (asiciy c, and h roorionaiy cofficin b ar considrd o hav consan vaus. g, fid imir is ocad a a disanc (Figur.; du o his, h sysm is cad on-sidd. osiiv vau of corrsonds o h imir osiion rsnd in Figur ; on h ohr hand, h imir has a ngaiv vau whn h sring is comrssd. Th osciaor moion is riodic, i.. im inrvas bwn wo conscuiv imacs ar h sam (idnica. Fig. mahmaica (dynamic mod of h g-mass on-sidd imac osciaor. Th diffrnia quaion of h sysm moion a h im inrvas bwn wo conscuiv imacs has h form of ( or (: m b c F ( 0 30 TRNSTIONS OF FMEN XLI-3 (07

3 naysis of Vibro-Imac Procsss of a Sing-Mass Sysm wih Viscous aming and a Sing Limir Lj. Garić n P ( Whr: n b / m is h viscous daming cofficin, c / m is h anguar frquncy of fr osciaions, P F / m is h raio of h rna corciv forc amiud and h sysm mass, 0 is h anguar frquncy of h rna corciv forc, is h iniia has of h rna corciv forc. I is known from h diffrnia quaion hory ha h souion of h homognous ar of quaion ( has h form of: h n n n (3 n or whr:, n is h anguar frquncy of damd osciaions. Th aricuar souion of h diffrnia quaion ( is akn in h foowing form:, whr and ar unknown ans. fr a fw comuing oraions, on can obain: P 4 n ; g n Th gnra souion of h inar inhomognous diffrnia quaion of h scond ordr, wih consan cofficins (, is obaind in h foowing form: h, or: (4 n n n (5 Unknown ingraion consans and ar usuay obaind dnding on h known iniia moion condiions. Sinc i is no ossib o drmin in advanc h iniia condiions undr which h osciaor moion wi hav a riodic characr, hs consans coud no b drmind in such a way. Insad of iniia moion condiions, condiions of h moion riodiciy wr usd for h drminaion of consans and ; hs condiions ar dfind in accordanc wih considrd mod of riodic osciaory sysm and ar givn by rssions (0 and ( in h coninuaion of his rsnaion. I is ncssary o drmin wha kind of raion bwn osciaor aramrs and aramrs of h rna corciv forc can rsu in riodic moions in which h riod of h osciaor imac a h imir is qua or rooriona o h riod of h rna corciv forc. In h rsnd anaysis i is assumd ha h duraion of h imac a h imir is oo shor in comarision wih h riod of h osciaor moion bwn wo conscuiv imacs; hrfor, i is ngcd in h cacuaions. Th ffc of h imac is akn ino accoun by h rsiuion cofficin 0 R. For h asic imac, R 0, whi for h absouy asic imac i is on, R. Taking TRNSTIONS OF FMEN XLI-3 (07 3

4 Lj. Garić naysis of Vibro-Imac Procsss of a Sing-Mass Sysm wih Viscous aming and a Sing Limir ino accoun h aformniond assumions, i is sufficin o obsrv ony on inrva of h osciaor moion bwn wo conscuiv imacs. On imac rrsns ony a ar of h conscuiv imac rocss, and h condiions undr which any of imacs occurs dnd on h condiions undr which h rvious imac has bn rformd. Procsss consig of an infini numbr of imacs ar cad infini imacs. Imacs ha ar rad ar characrisic of sysms wih riodic ciing forcs; his mans ha infini imac rocsss hav a riodic characr [], []. In hs sysms, h osciaor moion riod dnds on h riod of h rna corciv forc. Usuay, hr iss an osciaor moion mod wih h riod which is qua o h rna corciv forc riod, bu i is aso ossib ha hr is moion mods wih riods which ar rooriona o h rna corciv forc riod. This ossibiiy of various (mui mod isnc is h ssnia characrisic of vibro-imac sysms. u o his, h rsarch ino h vibro-imac sysm dynamics has o invov h dfiniion of a ossib ys of moion as w as h anaysis of h moion sabiiy. This shoud nab us o rica and rcogniz moion mods ha can is (occur in racic. In h rsnd rsarch, a g moion cyc bwn wo conscuiv imacs is considrd. Thrby, h osciaor vociis bfor and afr h imac ar diffrn: is h osciaor vociy bfor h imac, whi is h osciaor vociy afr h imac. Th raion conncing hs wo vociis is: R (6 3. ondiions for (aras of h isnc of riodic vibro-imac mods (odd, =,3, and vn, =,4, whn R 0. 7 and 0 riodic moion sysm is considrd and invsigad in his ar. I is assumd ha h osciaing riod is qua or rooriona o h rna corciv forc riod. In fac, his mans ha imacs occur a consan im inrvas, T /, or in roorion o h rna corciv forc riod, /, whr is h mod muiiciy. Whn =, hn, for a g fu riod of h rna corciv forc F( variaion, a g imac occurs a h fid imir. If =, hn, for wo fu riods T of h rna corciv forc F( variaion, a g imac a h imir occurs again, c. On moion cyc is obsrvd, bginning immdiay afr h imac; h im is masurd from ha momn uni h n imac. Whn nrgy osss in im inrvas bwn wo conscuiv imacs ar ngcd (which is usuay don, h diffrnia quaion of h osciaor moion can b rssd in h form of ( or (. If a aricuar souion is asummd in h form M N, afr h firs drivaiv drminaion and subsiuion in (, on can obain: P P n (7 4n 4n 3 TRNSTIONS OF FMEN XLI-3 (07

5 naysis of Vibro-Imac Procsss of a Sing-Mass Sysm Lj. Garić wih Viscous aming and a Sing Limir Th gnra souion of quaion (, in his cas, has h foowing form: X F X F n n n ( ( ( (8 whr: c F X / 0 is h saic disacmn of h mass undr h infunc of h forc 0 F, m b is h daming dcrmn, is h dimnsionss frquncy, and (,, F n m c. Inroducing racmns 0 ( X F and 0 ( X F ino rssion (8, on obains: 0 ( n n n (9 whr: ( F X, 0 g. For h considrd riodic moion of h sysm, boundary condiions ar assumd as foows: a for h bginning of h cyc: R 0 (0 (0 b for h nd of h cyc (i.. for h momn rcding h n imac: ( Sinc h vibro-imac mod is dfind in his mannr, i is sufficin o or a g imac cyc a h inrva ; 0. ondiions (0 and ( ar adod basd on h fac ha vociis rcding h imac hav h sam vaus a momns of im 0 and, i.. 0. Howvr, in ordr o raiz his mod, i is ncssary o us h corrsonding rna corciv forc ( F, i.. i is ncssary o choos h iniia has of h rna corciv forc, which aows h raizaion of h scd mod. TRNSTIONS OF FMEN XLI-3 (07 33

6 Lj. Garić naysis of Vibro-Imac Procsss of a Sing-Mass Sysm wih Viscous aming and a Sing Limir In racic, riodic mods wih a g imac of h mass a h imir during on riod of h rna corciv forc ar mosy usd. Basic characrisics of vibro-imac sysms ar mos cary rssd wih raivy sma vaus of h viscous daming cofficin and h daming dcrmn. Basd on h rssion: n n ( i can b adod ha. In addiion, if i is assumd ha: 0, hn rssion (9 can b wrin as: n ( (3 Th firs im drivaiv of rssion (3 rrsns h vociy: n n n (4 Now, i is ncssary o drmin cofficins and. Whn h boundary condiions (0 and ( ar inroducd ino rssions (3 and (4, afr a fw ransformaions, on can obain: С С С (5 B (6 R (7 R B (8 34 TRNSTIONS OF FMEN XLI-3 (07

7 naysis of Vibro-Imac Procsss of a Sing-Mass Sysm wih Viscous aming and a Sing Limir Lj. Garić Whrby, afr crain oraions and ransformaions in quaion (6, h vociy is obaind as: B (9 B Basd on rssion (9, i is ncssary o or h aras of vibro-imac mod isnc. Primariy, hr is a nd o drmin h ara in which h vociy has ra vaus. If i is assumd ha h rssion undr h squar roo in quaion (9 is osiiv, hn / B 0 h ara in which h ra vaus of h vociy ar ocad:. From his, on can obain h condiion which drmins B (0 ccording o raion (0, vibro-imac mods can is in h ara in which h imir is acd wihin h boundaris of h osciaion amiud of h inar osciaor 0, bu hy can aso is ousid ha ara:. In h ar cas, whn, hr is a nd o rovid addiiona nrgy for h vibro-imac mod. If condiion (0 is no saisfid, vibro-imac mods ar no ossib. I is obvious ha raion (9 is an ambiguous funcion of h sysm aramrs; hus, h characrisic of vibro-imac osciaory sysms is h occurnc of ambiguous souions. In addiion, anohr condiion is inroducd: h moion rajcory dos no cd h imir a h inrva bwn wo conscuiv imacs: ( ( In a gnra cas, i is no ossib o drmin h aras of vibro-imac mod isnc basd on condiion (. For sysms wih a raivy sim srucur, ony aroima bordrs of hs aras can b drmind. To do ha, i is rimariy imoran o saisfy h foowing condiion: 0 ( This condiion is no aid wihin boundary condiions and nds o b aid as an addiiona condiion in h moion quaion. L condiion (0 b saisfid; i is ncssary o drmin wha kind of rsricions occur du o h nd ha vociy has a osiiv vau. From raion (9 i foows ha if h rssion undr h squar roo is grar han, hn h vociy has wo diffrn vaus, osiiv and ngaiv. u o his, condiion ( is saisfid ony for a g vau of h vociy. onsquny, by obsrving quaion (9, i is ncssary o drmin whn h vociy is osiiv, i.. whn condiion ( is saisfid. In ha sns, in rssion (9, a sign of shoud b considrd a firs. Bcaus is givn by rssion (7, boh h numraor and h dnominaor of hav o b obsrvd. Th numraor is grar han zro (i.. osiiv, whn 0, and his is for h inrva k ; k. This yids: TRNSTIONS OF FMEN XLI-3 (07 35

8 Lj. Garić naysis of Vibro-Imac Procsss of a Sing-Mass Sysm wih Viscous aming and a Sing Limir k k ; k ( k / : ; k k ; h rciroca is k k (3 Th numraor is ss han zro (i.. ngaiv, whn 0, and his is for h inrva k; k. This yids: k k ; ( k ( k / :, k k ; h rciroca is k k Paricuar anion shoud b aid o h dnominaor in rssion (7. If boh h numraor and h dnominaor hav h sam signs, hn h raio / is aways osiiv. u o a sma vau of h daming dcrmn, vaus and (4 in (7 ar aroimay qua o. Th rssion in h dnominaor in (7 is 0: is vau is ss han zro hroughou h who inrva of variab. This assumion is rovn by h diagram of h dnominaor funcion in rssion (7: f ( (Figur, a and b. This diagram is obaind ug h rograms Wofram Mahmaica 7 (Figur a and Mahcad 4 (Figur b, whn and 0.. I can b sn from his diagram ha whn, h vau of f ( nds o h -ais, bu i dos no cu i, whi h 0;. dnominaor is ngaiv in h who inrva (a (b Fig. a, b Th diagram of h dnominaor funcion f ( from rssion (7. 36 TRNSTIONS OF FMEN XLI-3 (07

9 naysis of Vibro-Imac Procsss of a Sing-Mass Sysm wih Viscous aming and a Sing Limir Lj. Garić 3. Mod I Th anaysis sars wih h mod I which corrsonds o h osiiv sign in fron of h squar roo in rssion (9. Th iniia vociy, in his cas, is aways osiiv in ara (0, vn whn is / 0 in rssion (9. In his cas, h condiion ha h rssion undr h squar roo has o b grar han or qua o zro nds o b saisfid so ha h vociy coud hav ra vaus: / B 0 (5 fr a fw sim cacuaions, h foowing is obaind: B. B If h rciroca is akn, h sign changs again: B, i.., and afr h roo oraion, h foowing is obaind: B B. B Th ar raion rrsns rviousy rsnd rssion (0, which is h firs condiion drmining h ara in which ra vaus of h vociy ar ocad. Now, i is ncssary o or h scond condiion, which is rsnd by quaion ( and which dfins h condiion ha h vociy has o b osiiv, 0. u o h fac ha, for his mod, h sign in fron of h squar roo in (9 has o b osiiv, i is ncssary o obsrv h sign of in quaion (7. Bcaus h rssion for nds o b osiiv (consquny, h vociy is aso osiiv, i is ncssary ha boh h numraor and h dnominaor hav h sam signs (osiiv or ngaiv. Th dnominaor in is aways ngaiv, f ( 0, which is shown by Figurs a, b. u o his, h numraor in has o b ngaiv oo bcaus, in ha way, h rssion for has a osiiv vau. Prviousy i has bn shown ha h numraor is ngaiv whn 0 ; in ha cas h inrva, (4, was obaind. Thn, h foowing is vaid: k k ( / 0, i.. 0. This mans ha inrva (4 is h condiion for h mod I isnc, whi i has arady bn shown ha rssion (0 is h condiion for h isnc of h sam mod. Thrfor, in his way, h vibro-imac mod I is dscribd, and i is dfind by h osiiv sign in fron of h squar roo in rssion (9. In addiion, from inrva (4, on obains ha for, 4 for, and 3 6 for 3, c. This is shown by Figurs 3a, b, and c which rrsn h aras whr vibro-imacs is for h abov isd vaus of aramr. 3. Mod II Th mod II which corrsonds o h ngaiv sign in fron of h squar roo in rssion (9 is aso ord. TRNSTIONS OF FMEN XLI-3 (07 37

10 Lj. Garić naysis of Vibro-Imac Procsss of a Sing-Mass Sysm wih Viscous aming and a Sing Limir Firs, h cas whn / 0, which is h condiion for h vociy bing osiiv, is ord. In his mod, on aks h ngaiv sign in fron of h squar roo in (9 (for h mod I, his sign was osiiv. Thus, h rssion undr h squar roo has o b ss han bcaus, in ha cas, h numraor in rssion (9 is osiiv: / B (6 fr a fw cacuaions, i bcoms / souion If h rciroca is akn, h sign changs again: /,. This ads o h. This souion dos no saisfy h condiion (, (, and hus, i is no suiab for furhr invsigaion bcaus, in ha cas, a mod wihou an imac can is, bu aso a mod wih an imac (Figurs 3 a, b, c. Th mod wih an imac can is ony whn addiiona nrgy is divrd o h sysm, i.. whn h amiud of h rna corciv forc F ( is incrasd. Thus, h vibro-imac mod II is dscribd; i is dfind by h ngaiv sign in fron of h squar roo in rssion ( Mod III as, h mod III, which corrsonds o a ngaiv sign in fron of h squar roo in rssion (9, is ord. Howvr, unik h mod II, hr h cas / 0 has o b ord (for mod II, i is h cas / 0 so ha h vociy is osiiv. In accordanc wih his fac, h rssion undr h squar roo in (9 has o b grar han, which is in conras wih h mod II. u o his, h numraor in rssion (9 has o b ngaiv. Thrfor, if h vociy is o b osiiv, i is rquird ha / 0, i.. 0. In ohr words, in his cas, boh h numraor and h dnominaor in rssion (9 ar ngaiv, and hus, h vociy is osiiv. I has bn shown rviousy ha is ngaiv whn h numraor in (7 is osiiv, bcaus is dnominaor f ( is aways ngaiv, and his hods for inrva (3,. k k Hnc, in ordr o hav a osiiv vociy in mod III, i is ncssary o or h cas whn h rssion undr h squar roo in (9 is grar han. In his cas, from (9 on can obain h foowing rssion: / B (7 fr a fw cacuaions, i bcoms: / If h rciroca is akn, h sign changs again:,. Thus, boh condiions ar saisfid: and h vociy is osiiv. In addiion, from inrva (3, on obains for, hn 4 for and 6 for 3. This is shown by Figurs 3 a, b, c which rrsns h aras in which vibro-imacs is for h abov isd vaus of aramr. Basd on h anaysis of h rsnd hr mods, i can b concudd ha in h frquncy inrva, for a ra vaus of aramrs which saisfy h condiion, vibro-imac mods can is. For, h vociy is qua o zro, which corrsonds o h osciaor moion, nding ony wih h ouch of h imir (wihou imac. Th conducd rsarch has shown ha vibro-imac mods can is vn in h cas whn. 38 TRNSTIONS OF FMEN XLI-3 (07

11 naysis of Vibro-Imac Procsss of a Sing-Mass Sysm wih Viscous aming and a Sing Limir Lj. Garić (a (b (c Fig. 3 a, b, c ras of vibro-imac mod I, II, and III isnc. Figurs 3 a, b, c show h aras of vibro-imac mod I, II, and III is. iagrams ar od by ug sofwar rogramms Mahcad 4 and Wofram Mahmaica 7, ug rssion (0, for h consan vaus of R and, and for h corrsonding vaus of TRNSTIONS OF FMEN XLI-3 (07 39

12 Lj. Garić naysis of Vibro-Imac Procsss of a Sing-Mass Sysm wih Viscous aming and a Sing Limir ( =,, and 3. Ths figurs cary show ha, by incrag muiiciy, aras of mod isnc incras, oo, i.. bcom widr. iagrams in Figurs 3a, b, c nab us o drmin and dfin h frquncy inrva of h vibro-imac rocss, whn h vau of h disanc is known. Ths diagrams aso show vibro-imac mod isnc, whn h disanc is grar han h osciaion amiud,, (mod II. In addiion, rssions for h dnominaor f ( of quaion (7 ar rsnd wih corrsonding diagrams drawn by h mahmaic ackag Wofram Mahmaica 7 (Figurs 4 a, b, c.. h nd of h comuaion, comy h sam diagrams as in Figurs 3 a, b, c ar obaind for h aras whr vibro-imac mods is whn,, and 3. (a (b 40 TRNSTIONS OF FMEN XLI-3 (07

13 naysis of Vibro-Imac Procsss of a Sing-Mass Sysm wih Viscous aming and a Sing Limir Lj. Garić (c Fig. 4 a, b, c ras of vibro-imac mod I, II, and III isnc. Figur 5 shows a has diagram drawn by Mahcad 4 for wo diffrn vaus of h daming dcrmn, 0. and 0., whn h foowing aramrs ar known: F0 N, X N, m kg, s, m, R 0. 7,. 5,.5s. Lin of imac Fig. 5 Phas orrai of h im inrva T bwn wo conscuiv imacs. 4. ondiions for (aras of h riodica vibro-imac mod isnc (odd, =,3, and vn, =,4, whn R and 0. Nw (addiiona rsus, giving mor informaion abou h aras in which vibro-imac mods is, ar vry significan. In ha sns, a cas of h absouy asic imac ( R, wih a daming dcrmn of 0, is ord. In his cas, h foowing is obaind from raions (7 and (8: TRNSTIONS OF FMEN XLI-3 (07 4

14 Lj. Garić naysis of Vibro-Imac Procsss of a Sing-Mass Sysm wih Viscous aming and a Sing Limir 0 B (8 arrying ou crain oraions and ransformaions in (9, h vociy bcoms: B B... / (9 fr corrsonding subsiuions in (3, h foowing aw of h sysm moion is obaind: ( (30 Th rm vau is obaind for:, i.. whn: 0, from whr h foowing is obaind:, i... I is ncssary o saisfy condiion (, and, in ha way, rssion (30 bcoms: (3 Th rssion for h vociy (9 is subsiud in quaion (6, rsuing in: Finay, on obains h foowing: (3 Taking ino accoun rssion (3, afr corrsonding ransformaions in inquaiy (3, h foowing condiions ar obaind: a Whn is vn ( =, 4, 6, : ; 4 TRNSTIONS OF FMEN XLI-3 (07

15 naysis of Vibro-Imac Procsss of a Sing-Mass Sysm wih Viscous aming and a Sing Limir Lj. Garić Finay, on obains:, whn i is : and, whn i is : b Whn is odd ( =, 3, 5, : Finay, h foowing is obaind: cg, whn i is : 0 and cg, whn i is : 0 In h obsrvd and ord vibro-imac rocss, h osciaion cnr movs du o asymmric on-sidd imacs. Th osciaion cnr is markd by h oin O (Figur, and h osciaor moion shoud b h sam (symmric on boh sids (f and righ of h oin O. Howvr, osciaions ar no symmric bcaus h mass imacs h imir on h righ sid, whi, on h f sid, h mass movs wihou any imac. 5. oncusion naysis of h horizona sraigh-in moion of a g-mass, on-sidd (wih a g imir vibro-imac osciaor is rsnd in his ar. Th osciaor moion is riodic, wih h consan riod T, which rrsns h im inrva bwn wo conscuiv imacs. Th forciby damd imac osciaor is osd o h acion of h foowing facors: an asic forc F, a viscous daming forc F w, and an rna riodic corciv forc F (. Two cass ar ord: h firs cas, whn h rsiuion cofficin is R 0.7 and h daming dcrmn is 0, and h scond cas, whn R and 0. In h firs cas, h aras of riodica vibro-imac mod isnc ar ord, whn h mod muiiciy has boh vn and odd vaus. Th obaind rsus ar rsnd in Figur 3 (a, b, and c, showing h isnc of hr vibro-imac aras. I can b cary sn ha, in a gnra cas, whn h mod muiiciy has a grar vau, h aras of vibroimac mod isnc ar widr. In h scond cas, h aras of riodica vibro-imac mod isnc, whn h mod muiiciy has boh vn and odd vaus, ar ord and h obaind rsus ar rsnd by corrsonding rssions. For h cacuaion and grahic visuaizaion of h vibro-imac sysm dynamics, mahmaica sofwar ackags Мahcad 4 and Wofram Mahmaica 7 wr usd. This ar rsns h rsus obaind for a dfind form of h vibro-imac sysm srucur (wih a g-mass, a g imir and on sring, which is osd o h infunc of a riodic rna forc. Nvrhss, h obaind rsus can giv som idas and guidoss for rsarch ino vibro-imac sysms wih diffrn srucurs and diffrn ys of rna forc. ; (33 (34 TRNSTIONS OF FMEN XLI-3 (07 43

16 Lj. Garić naysis of Vibro-Imac Procsss of a Sing-Mass Sysm wih Viscous aming and a Sing Limir Th main rsu and h concusion of h rformd rsarch is h foowing: whn h viscous daming cofficin has a smar vau, h aras of vibro-imac mod isnc bcom widr. This mans ha daming dcrass h aras in which vibro-imac mods is, bu i can no caus h oa disaaranc of vibro-imacs. Whn chnica (working condiions aow, h fficin souion o onia robms is o incras h disanc from h imir byond h boundaris of isnc of a givn vibro-imac mod. Such a srucur is raizd on h basis of amiud-frquncy characrisics of h vibro-imac sysm rsnd by Figur 3 (a, b, and c. REFERENES [] Babicki V. I., Thory of Vibro-Imac Sysms and icaions, Sringr-Vrag, Brin, 998. (Rvisd ransaion on Engish from Russian, Nauka, Moscow, [] Jakovvic M. I., Emni orii oimanogo uravnija riodičskimi ržimami vibroudarnih sism: Sismi, injni v romžukah mždu soudarniami, Moskva, 00. [3] Bards. E., Srucura Vibraion: naysis and aming, Univrsiy of London, 996. [4] Gndman O.V., oni., ynamics of forcd sysm wih vibro-imac nrgy k, Journa of Sound and Vibraion, Voum 358, 05, Pags hs://doi.org/0.06/j.jsv [5] M. Xu, Y. Wang, X. L. Jin, Z. L. Huang, T. X. Yu, Random rsons of vibro-imac sysms wih inasic conac, Inrnaiona Journa of Non-Linar Mchanics, 03. hs://doi.org/0.06/j.ijnoninmc [6] Jingyu Wang, Haoian Wang, and Ti Wang, Erna Priodic Forc onro of a Sing-gr-of- Frdom Vibroimac Sysm, Journa of onro Scinc and Enginring, Voum 03. [7] Prchikov N., Gndman O.V., ynamics and sabiiy of a discr brahr in a harmonicay cid chain wih vibro-imac on-si onia, Physica : Noninar Phnomna, 05, Voums 9, 8-8. hs://doi.org/0.06/j.hysd [8] Kmbr S.., Babickij V.I., Eciaion of vibro imac sysm by riodic imuss, Journa of Sound and Vibraion; 999, Vo. 7. hs://doi.org/0.006/jsvi [9] Yang Liu, Ekarina Pavovskaia, Marian Wircigroch, ZhikPng, Forward and backward moion conro of a vibro-imac casu sysm, Inrnaiona Journa of Non-Linar Mchanics, 05, hs://doi.org/0.06/j.ijnoninmc [0] Liviu Brindu, Sabiiy of h riodic moions of h vibro-imac sysms, haos, Soions and Fracas, 000, Pags hs://doi.org/0.06/s ( [] Sivio L.T. d Souza, Ibr L. adas, onroing chaoic orbis in mchanica sysms wih imacs, haos, Soions and Fracas 9, 004, Pags hs://doi.org/0.06/s ( [] Jian Mao, Yu Fu, Pichao Li, ynamics of riodic imusiv coision in scamn mchanism, Shock and Vibraion 0, 03, Pags hs://doi.org/0.55/03/35049 Submid: ccd: Ljubiša Garić Taching assisan armn of Mchanica Enginring Facuy of Tchnica Scincs Univrsiy of Prišina 380 Kosovska Mirovica Knza Mioša 7 Fa/: jubisa.garic@r.ac.rs 44 TRNSTIONS OF FMEN XLI-3 (07