Self-excited Vibration of Ink Rollers

29 ACC JOURNAL 2021, Volume 27, Issue 1 DOI: 10.15240/ ul/004/2021-1-003 SELF-EXCITED VIBRATION OF INK ROLLERS Ma in Pus ka1; Pa el Šidlo 2 VÚTS, a.s., Measu emen Depa men , S á o ská 619, Libe ec XI – Růžodol I, 460 01 Libe ec, Czech Republic e-mail: 1[email p o ec ed]; 2pa e[email p o ec ed] Abs ac A ib a ion ha ing a cha ac e o sel -exci ed cha e oscilla ion known om machine ools is obse ed du ing in e mi en mo ion o ink olle s o o se p in ing machines. This ib a ion occu s unde speci ic ope a ing condi ions and is o en accompanied by an inc eased noise le el. To explain his unusual ib a ion beha io , a simple analy ical model o wo olle s in e ac ion is de i ed. The calcula ed oscilla ion is compa ed wi h he measu emen o duc o olle displacemen . The model esul s con i m he possibili y o sel -exci ed ib a ion de elopmen in he p esence o iscous o ces, nega i e damping e ec s and con inuous supply o ex e nal ene gy om olle o a ion. Keywo ds Sel -exci ed oscilla ion; Viscous ic ion o ce; Nega i e damping; O se p in ing machine. In oduc ion O se p in ing is widely used and cu en ly he mos impo an p in ing echnology. The con inuous de elopmen and au oma ion o o se p in ing machines p o ide inc eased p oduc i i y and highe media quali y. Howe e , he equi emen s o highe p oduc ion speed also inc ease he dynamic loading o mechanisms, o en accompanied by a signi ican noise and a ib a ion emission. The inking uni o o se p in ing machine ans e s he ink om he ink oun ain o he pla e cylinde [1]. To ha e a hin, uni o m ilm o ink, he inking uni is composed o a sys em o many ( en o mo e) olle s. The quan i y o ink supplied o he inking p ocess is de e mined by he adjus men and he speed o he inpu pa shown in Fig. 1. Sou ce: Own Fig. 1: Scheme o he inpu pa o inking uni INK FOUNTAIN ROLLER INK FOUNTAIN DUCTOR ROLLER OSCILLATOR ROLLER 30 The ink oun ain olle and he oscilla o olle (including duc o olle ipping mechanism) a e d i en independen ly and he olle s ha e subs an ially di e en ci cum e en ial eloci ies. The duc o olle o a es eely and is al e na ely in con ac wi h he oun ain olle and wi h he oscilla o olle o ans e a speci ied amoun o ink. A he momen o con ac , he duc o olle ab up ly changes i s o a ional speed. A high p oduc ion speeds, he olle impac s a e associa ed wi h an inc eased noise and a ib a ion o he duc o olle . These e ec s occu only unde speci ic ope a ing condi ions, i.e. o olle s co e ed by an ink laye and a a ce ain in e al o o a ional speed. The ib a ion wa e o m indica es he po en ial occu ence o a sel -exci ed oscilla ion. 1 Resea ch Objec i es The sel -induced ib a ion o ink olle s is an in e es ing and a he unusual phenomenon. The aim o he esea ch is an ini ial s udy o he causes and sou ces o his ib a ion beha io including posi i e and nega i e damping e ec s. A simpli ied analy ical model o he olle in e ac ion is c ea ed o app oxima ely simula e he measu ed olle ib a ion. 2 Vib a ion o Duc o Rolle An example o a ajec o y o duc o axis cen e poin in a plane pe pendicula o his axis wi hin one cycle o a ecip ocal mo ion be ween an ink oun ain and an oscilla o olle is depic ed in Fig. 2. The de lec ion o he axis cen e poin is de i ed om he displacemen o he duc o su ace measu ed by means o wo lase iangula ion senso s in he middle o he duc o leng h. This ajec o y is no exac ly he same in indi idual cycles due o a high sensi i i y o ini ial and bounda y condi ions. Sou ce: Own Fig. 2: Plane mo ion o duc o axis cen e poin (one cycle o he ecip ocal mo ion be ween ink oun ain and oscilla o olle ), ed – angen and no mal lines o he adjacen olle s a a s eady olling An excessi e ib a ion o duc o axis cen e poin (a bending de lec ion) is clea ly seen in bo h e e sal poin s, pa icula ly a he con ac wi h he oscilla o olle . The de ail o axis cen e poin displacemen (in he di ec ion 𝑥 o he angen o he oscilla o ) and he co esponding sound p essu e a he impac on he oscilla o a e shown in Fig. 3. The measu emen mic ophone was loca ed nea he olle su ace. 31 Sou ce: Own Fig. 3: The oscilla ing de lec ion o duc o axis cen e poin ( op) and sound p essu e nea duc o olle (bo om) a he momen o he impac on he oscilla o olle The ib a ion beha io shown in Fig. 3 is appa en ly non-linea . A e he con ac o olle ink laye s (app oxima ely a a ime o 0 s) he ib a ion ampli udes s a o inc ease g adually wi h a equency o duc o undamen al bending mode. This beha io is simila o he sel -exci ed cha e ib a ions o machine ools [2] o o o he mechanical o luid sys ems [3]. The sel - exci ed ib a ions a e gene ally cha ac e ized by a nega i e damping, a posi i e eedback and an ene gy impo om an ex e nal sou ce. A e a ce ain ime (app oxima ely a 0.06 s) he ib a ion ampli udes apidly educe, which co esponds o a common cha ac e is ic o a damped mechanical sys em wi h posi i e damping. To explain his ib a ion beha io , a simpli ied model o he olle in e ac ion is de i ed in he nex chap e . 3 Resea ch Me hods 3.1 Model o Rolle Vib a ion The ansien ib a ion o he duc o olle du ing he con ac o he duc o and he oscilla o is modelled using he analy ical model depic ed in Fig. 4. The model is based on a simpli ied kinema ic geome y o a eal sys em and on a numbe o o he simpli ying assump ions. The duc o olle is subs i u ed by a cylind ical disc o mass 𝑚 and momen o ine ia 𝐽. The disc axis is lexibly moun ed in he ho izon al di ec ion 𝑥, oscilla ions in he e ical di ec ion a e neglec ed. The alues o s i ness 𝑘 and iscous damping coe icien 𝑏 co espond o he eigen equency and modal damping o he olle undamen al bending mode [4]. The e ical displacemen o he disc is de ined by a cam mechanism li 𝑦 shown in Fig. 5. In he end posi ion he olle s a e comp essed and de o med ( he duc o has a ubbe su ace) and he con ac a ea be ween olle s (a olle nip) is 𝑆 = 𝑝0𝑙, whe e 𝑝0 is he s a ic nip wid h and 𝑙 is he disc leng h. Bo h olle s ha e he same ink laye o hickness ℎ and a he momen o impac 𝑡 = 0 hey a e sepa a ed by a gap o heigh 2ℎ. We suppose ha he ink (which is hixo opic in ac ) exhibi s he New onian low cha ac e is ics [5] and has a cons an dynamic iscosi y 𝜂. Be o e he impac , he oscilla o o a es wi h a cons an angula eloci y 𝜔0 and he angula eloci y 𝜔 o he disc is ze o. A he con ac o ink laye s ( ime 0 s in Fig. 5) he iscous 32 ic ion o ce 𝐹 is de eloped, which spins he model disc wi h he inc easing eloci y 𝜔 and simul aneously exci es he ib a ion in he ho izon al di ec ion. The o ce 𝐹 is p opo ional o he eloci y g adien o ink laye and o he momen a y nip wid h 𝑝. Sou ce: Own Fig. 4: Model o duc o olle ansien ib a ion a he olle con ac Sou ce: Own Fig. 5: Cam li 𝑦(𝑡) 3.2 Equa ions o Mo ion The go e ning equa ions o mo ion ha e a o m 𝑚 𝑥󰇘(𝑡)+ 𝑏 𝑥󰇗(𝑡)+ 𝑘 𝑥(𝑡)−𝐽 𝑟1 𝜔󰇗(𝑡) = 0, (1) 𝐽 𝑟1 𝜔󰇗(𝑡)− (𝑟2 𝜔0− 𝑟1 𝜔(𝑡)+ 𝑏𝑠(𝑡) 𝑥󰇗(𝑡))1 2ℎ 𝜂 𝑝(𝑡) 𝑙 = 0, (2) whe e 𝑟1, 𝑟2 a e duc o and oscilla o diame e s. The momen a y nip wid h 𝑝 sa is ies a simpli ied geome ical condi ion 𝑝(𝑡) = √−4 𝑟𝑠 𝑦(𝑡) − 𝑦2(𝑡) − 𝑥2(𝑡) o 𝑦(𝑡)< 0 and 𝑥2(𝑡)> 4 𝑟𝑠 𝑦(𝑡) + 𝑦2(𝑡), (3) 2h k y x F J,m 1 2  , 0  b DUCTOR ROLLER (DISC) OSCILLATOR ROLLER 33 𝑝(𝑡) = 0 elsewhe e, (4) whe e 𝑟𝑠=𝑟1 + 𝑟2 2 is he mean olle s adius. The a iable damping necessa y o he sel -exci ed ib a ion ollowed by he damped ib a ion (as discussed in In oduc ion) is modelled by he damping unc ion 𝑏𝑠, which is in oduced in o Eq. (2) o he iscous ic ion o ce. The unc ion 𝑏𝑠 shown in Fig. 6 was c ea ed a i icially using a c angen unc ion o co espond o he measu ed ime his o y in Fig. 3. Sou ce: Own Fig. 6: Damping unc ion 𝑏𝑠(𝑡) The model o olle ib a ion has wo deg ees o eedom, namely he ho izon al displacemen o duc o axis 𝑥 and he duc o angula eloci y 𝜔. The esul ing ime cou ses can be ob ained by a simul aneous solu ion o Eqs. (1) and (2) wi h ini ial condi ions 𝑥(0) = 0 and 𝜔(0) = 0. 4 Calcula ion Resul s The model pa ame e s used o calcula ion a e gi en in Table 1. These nume ic alues co espond o he dimensions and pa ame e s o a la ge- o ma shee ed o se p ess. Tab. 1: Calcula ion pa ame e s e ec i e mass 𝑚 [kg] 8.0 momen o ine ia 𝐽 [kg.m2] 0.0283 e ec i e s i ness 𝑘 [N/m] 6.8·106 iscous damping coe icien (𝜉 = 3%) 𝑏 [Nms-1] 440 duc o diame e 𝑟1 [m] 0.059 oscilla o diame e 𝑟2 [m] 0.076 e ec i e disc leng h 𝑙 [m] 0.35 s a ic nip wid h 𝑝0 [m] 0.003 ink hickness ℎ [m] 10-5 ink iscosi y 𝜂 [Pa.s] 10 oscilla o angula eloci y 𝜔0 [ ad/s] 78 Sou ce: Own The calcula ed ime cou ses o duc o axis displacemen 𝑥, angula eloci y 𝜔, di e ence eloci y Δ𝑣 and iscous o ce 𝐹 a e depic ed in Fig. 7. The di e ence o he oscilla o and he duc o ci cum e en ial eloci y Δ𝑣 is ob ained om 34 Δ𝑣(𝑡) = 𝑟2𝜔0− 𝑟1𝜔(𝑡), (5) he angen ial iscous o ce 𝐹 be ween olle s is gi en by 𝐹(𝑡) = 𝐽 𝑟1 𝜔󰇗(𝑡). (6) Sou ce: Own Fig. 7: Calcula ed ime cou ses o ( op o bo om) duc o disc axis ho izon al displacemen 𝑥(𝑡), angula eloci y 𝜔(𝑡), di e ence ci cum e en ial eloci y 𝛥𝑣(𝑡) and angen ial iscous o ce 𝐹(𝑡) The ime ma k a 0 s in Fig. 7 indica es he con ac o ink laye s (see also Fig. 5). The ime cou se o he axis ho izon al displacemen 𝑥 co esponds ela i ely well o he measu ed his o y shown in Fig. 3. The mos c ucial is he di e ence o displacemen and o ce ampli udes a he beginning, which is he esul o a sudden de elopmen o he iscous o ce 𝐹 a he olle s con ac . The model assumes a cons an alue o dynamic iscosi y 𝜂 and a cons an eloci y g adien . Howe e , he local ink iscosi y a he con ac su ace ab up ly dec eases due o he hea ing o a small amoun o ink caused by ic ion. Wi h a g adual inc ease o duc o speed inc eases also he low o adhe ed ink and he ink hea ing and iscosi y dec ease a e educed. The o ce ampli ude inc eases in ac g adually om ze o. 35 The e ec o a iable damping is clea ly isible in Fig. 7. A e a s abiliza ion o he angula eloci y 𝜔 he inc easing ib a ion is supp essed, and he ib a ion ampli udes apidly educe. F om his we can conclude ha a he beginning he duc o ib a ion is e iden ly sel -exci ed wi h he o a ing oscilla o olle being an ex e nal ene gy sou ce. When he di e ence ci cum e en ial eloci y 𝛥𝑣 signi ican ly dec eases, he ex e nal ene gy disappea s, and he damping o he sys em becomes posi i e. Conclusion An in e es ing phenomenon o a ib a ion simila o a sel -exci ed cha e oscilla ion was ecen ly obse ed on p in ing machines du ing an in e mi en mo ion o ink olle s. The excessi e ib a ion o duc o olle occu s only in he p esence o ink laye a a ce ain in e al o o a ional speed and is accompanied by a high noise le el. To cla i y he o igin o his ib a ion unde qui e complica ed condi ions, a simpli ied analy ical model o olle s con ac was c ea ed. The model esul s indica e a p obable sou ce o he ib a ion beha io , which is a iscous ic ion o ce o igina ing in he ink laye be ween olle s. In he p esence o an ene gy impo caused by di e en ci cum e en ial eloci ies o bo h olle s he ink laye in e ac ion c ea es a posi i e eedback, which exci es he sel - oscilla ion o he duc o olle . The exci a ion e ec o he iscous o ce apidly disappea s a e he eloci y di e ence signi ican ly dec eases. The simpli ied model desc ibes his ib a ion p oblem only app oxima ely and he p ocess o olle sel -exci a ion should be u he analyzed, e.g., by means o ad anced modelling using FEM and compu a ional luid dynamics me hods. Acknowledgemen s This wo k was suppo ed by he Czech Minis y o Indus y and T ade in he amewo k o he ins i u ional suppo o long- e m concep ual de elopmen o esea ch o ganiza ion whose ecipien was VÚTS, a.s. Li e a u e [1] KIPPHAN, H. (ed.): Handbook o P in Media: Technologies and P oduc ion Me hods. Sp inge -Ve lag Be lin Heidelbe g, 2001. ISBN 978-3-540-67326-2. eBook ISBN 978- 3-540-29900-4. DOI: 10.1007/978-3-540-29900-4 [2] QUINTANA, G.; CIURANA, J.: Cha e in machining p ocesses: A e iew. In e na ional Jou nal o Machine Tools and Manu ac u e. 2011, Vol. 51, Issue 5, pp. 363–376. DOI: 10.1016/j.ijmach ools.2011.01.001 [3] TONDL, A.: To he p oblem o sel -exci ed ib a ion supp ession. Enginee ing Mechanics. 2008, Vol. 15, Issue 4, pp. 297–307. A ailable om WWW: h p://www.enginee ingmechanics.cz/pd /15_4_297.pd [4] BREPTA, R.; PŮST, L.; TUREK, F.: Mechanické kmi ání. Sobo áles, P aha, 1994. ISBN-13 978-80-901684-8-0. ISBN 80-901684-8-5. [5] CHHABRA, R. P.: Non-New onian Fluids: An In oduc ion. In: K ishnan J.; Deshpande A.; Kuma P. (eds), Rheology o Complex Fluids. Sp inge , New Yo k, NY, 2010, pp. 3–34. ISBN 978-1-4419-6493-9. Online ISBN 978-1-4419-6494-6. DOI: 10.1007/978-1-4419-6494-6_1 doc. Ing. Ma in Pus ka, Ph.D.; Ing. Pa el Šidlo , CSc. 36 SAMOBUZENÉ KMITÁNÍ BARVICÍCH VÁLCŮ Při s řída ém pohybu ba icích álců isko ého s oje se ysky ují ib ace mající cha ak e samobuzených kmi ů, k e é jsou známy například u ob áběcích s ojů. Ty o ib ace znikají při u či ých p o ozních podmínkách a jsou čas o dop o ázeny z ýšenou hladinou hluku. P o objasnění oho o neob yklého kmi a ého cho ání je od ozen jednoduchý analy ický model zájemného působení d ou álců. Vypoč ené ib ace jsou po o nány s měřením posunu í ba icího álce. Výsledky modelo ání po zují možnos y olání samobuzených kmi ů při pří omnos i iskózních sil, účinků zápo ného lumení a pří odu nější ene gie ořené o áčením álce. SELBSTERREGTE SCHWINGUNG DER FARBWALZEN Bei de in e mi ie enden Bewegung de Fa bwalzen i eine Schwingung au , die den Cha ak e eine selbs e zeug en Schwingung ha und die aus dem Be eich de We kzeugmaschinen bekann is . Diese Schwingung en s eh un e bes imm en Be iebsbedingungen und wi d o on Ge äuschemission beglei e . Fü die E klä ung dieses ungewöhnlichen Schwing e hal ens wi d das ein ache analy ische Modell de Zwei-Walzen- Gegenwi kung en wo en. Die be echne e Schwingung wi d mi de Messung des Wegs on Fa bwalzen e glichen. Die Modelle gebnisse bes ä igen die Möglichkei , eine selbs e zeug e Schwingung in Gegenwa on Viskosk ä en, on Wi kungen nega i e Dämp ung und de Zu üh ung de äuße en Ene gie de Walzend ehung zu e zeugen. DRGANIA SAMOWZBUDNE WAŁKÓW FARBOWYCH Podczas p zemiennego uchu wałków a bowych maszyny d uka skiej wys ępują d gania mające cha ak e d gań samowzbudnych, k ó e wys ępują p zykładowo w maszynach do ob óbki. D gania e pows ają w pewnych wa unkach eksploa acyjnych i częs o owa zyszy im wyższy poziom hałasu. Dla wyjaśnienia ych nie ypowych d gań op acowano p os y model anali yczny wzajemnego oddziaływania dwóch wałków. Obliczone d gania po ównano z pomia ami p zesunięć wałka a bowego. Wyniki modelowania po wie dzają możliwość wywołania d gań samowzbudnych w obecności opo u wisko ycznego, oddziaływania opo u ujemnego i wpływu ene gii zewnę znej gene owanej p zez ob o y wałka.