Vibration damping of the cantilever beam with the use of the parametric excitation
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1 The s Ieraioal Cogress o Soud ad Vibraio 3-7 Jul, 4, Beijig/Chia Vibraio dampig of he cailever beam wih he use of he parameric exciaio Jiří TŮMA, Pavel ŠURÁNE, Miroslav MAHDA VSB Techical Uiversi of Osrava Czech Republic
2 Czech Republic - Osrava ICSV 4
3 Oulie Moivaio Parameric resoaces Damped Mahieu equaio umped parameer model of he cailever beam Malab-Simulik model ocaio of he pach piezoacuaor a he clamp of he beam Effec of he exciaio ampliude o he deca rae of he free ed Effec of he radom disurbig force o dampig Coclusios 3 ICSV 4
4 Moivaio Ho. Prof. Dr. Ig. Aleš Todl DrSc., Dr. h.c. Hisor frequec Simulaio resul (obaied wih a aalog compuer) of a self-excied ssem exhibiig vibraio suppressio ear he parameric combiaio resoace frequec η = Ω Ω. From [Todl, 998] New papers Hors Ecker: Parameri exciaio i egieerig ssems, h Ieraioal Cogress of Mechaical Egieerig, November 5-, 9, Gramado, RS, Brazil B. Peermeier ad H. Ecker, Vibraio suppressio of a cailever beam b ope loop corol of a aached siffess eleme. Proceedigs of ENOC 8, Sai Peersburg, Russia, 3 Jue, 4 Jul, 8. Peermeier s ad Ecker s paper uses he beam model of he Timosheko pe. 4 ICSV 4
5 Parameric resoaces Priciple Parameric Resoaces a frequecies Pr j j,,,... Combiaio Parameric Resoaces a frequecies Cr j k j k, j, k,,...,,,... are he j-h ad k-h aural j ad frequec of he liear ssem (µ = ). Time-varig siffess, cos are parameers Problem of sabili iear ime-ivaria (TI) ssem? 5 ICSV 4
6 Damped Mahieu equaio d d d d Priciple Parameric Resoaces a frequecies cos Sabili regios Pr,,,... is a aural frequec of he liear ssem. iear par of he equaio d d d d No-saioar feedback cos iear par of he equaio cos 6 ICSV 4
7 umped parameer model of he cailever beam 7 ICSV 4. N N V : : :. cos cos.,,,,,, T e F,, M C F C M e Poeial eerg V of he defleced beam is as follows.,...,,, d d N V T T agrage's equaios of moio Variable siffess affecs ol 3 of N equaios where Periodicall varig siffess is replaced b a periodic exeral forces whose ampliude is proporioal o he deflecios of 3 elemes , B A A B A A B M. 3, 3 4 h m B h m A umped parameer model Euler-Berouli beam
8 Malab-Simulik model Malab-Simulik model of he beam for five elemes ad feedbacks Malab-Simulik lumped parameer model of he cailever beam for arbirar umber of elemes iear ime-ivaria (TI) ssem Feedback Idex k [Hz] [rad/s] Ceral differece of he secod order z 8 ICSV 4
9 ocaio of he pach piezoacuaor a he clamp of he beam Effec of he pach piezoacuaor posiio o he dampig of vibraios Simulik model Feedback Aleraive liear piezoacuaor 9 ICSV 4
10 Effec of he exciaio ampliude o he deca rae of he free ed Iiial beam deformaio Effec of he μ o he ime of deca, a which he vibraios are reduced b 4 db ( imes) Cr 546 rad/s Sead sae effec Deca of he free ed of he beam wihou parameric dampig Effec of he ime varig siffess Evelope i db db log 5 log 5 ICSV 4
11 Effec of he radom disurbig force o dampig Frequec specrum of he exciaio ad respose a he beam free ed for f Cr Cr f f f, 86 9 Hz Cr f f f f airesoace frequec Spliig of he domia specrum peak io wo smaller peaks ICSV 4
12 Coclusio o o o o The objecive of his paper was o demosrae ha parameric dampig reduces vibraio. Ampliude chages i siffess were chose b he simulaio approach. The parameric exciaio is oe of he ools o icrease he efficiec of vibraio dampig. The paper examied he Priciple ad Combiaio Parameric Resoace frequecies ad heir effec o he deca rae for he cailever beam. I was foud ha he greaes effec o vibraio dampig has he differece frequec bewee he firs ad secod resoace frequec of he cailever beam. This frequec differece is a Combiaio Parameric Resoace frequec of he firs order. The opimum size of half he ampliude of exciaio for his frequec was also deermied. Oher parameric resoace frequecies are wihou effec o he vibraio dampig. The frequec specra clearl explai wh icreases dampig of he paramericall damped ssems. The domiaig peak i he specrum splis io wo adjace peaks ad heir magiude is reduced ICSV 4
13 Thak ou for aeio hp://homel.vsb.cz/~um5 ICSV 4 3
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