VIBRATION DAMPING OF THE CANTILEVER BEAM WITH THE USE OF THE PARAMETRIC EXCITATION

Transcription

1 The st Iteratioal Cogress o Soud ad Vibratio -7 Jul eijig/chia VIRTIO DMPIG OF THE CTIEVER EM WITH THE USE OF THE PRMETRIC EXCITTIO Jiří Tůma Pavel Šuráek Miroslav Mahdal VS Techical Uiversit of Ostrava Ostrava Czech Republic jiri.tuma@vsb.cz pavel.suraek@vsb.cz miroslav.mahdal@vsb.cz The effect of the parametric vibratio dampig is theoreticall well described ad the first eperimets are published. The paper deals with the aalsis of the parametric ecitatio of the vibratio bod ad the catilever beam which is modeled b the lumped parameter sstem. The priciple of the parametric ecitatio is based o the use of piezoactuators as a elemet of the cotrolled stiffess which varies periodicall i time accordig to a siusoidal fuctio. The ecitatio frequec is selected the frequec of the priciple parametric resoaces ad combiatio parametric resoaces. The mai topic of the paper is focused o the aalsis of the effect of these frequecies ad the amplitude of ecitatio o the dampig ratio. The paper presets a simulatio stud of this method of dampig vibratios.. Itroductio catilever beam is a simple eample of a mechaical structure that ca serve as a object for testig the active vibratio cotrol. promisig wa to reduce vibratio of mechaical structures is parametric ecitatio which is fudametall differet from the active vibratio cotrol sstems developed b Krek []. Ulike active vibratio dampig which is based o the use of liear methods for cotrol of the liear time ivariat sstems (TI) the ew wa of dampig uses a periodic chage of a parameter usuall sprig stiffess K as it is show i Fig.. Of course the mea value of the stiffess K is costat i time. Such a sstem becomes o-liear ad potetiall ustable for a iterval of the frequec ad of the metioed parameter variatio. Figure. Parametric ecitatio of a liear time-ivariat (TI) multibod sstem. fudametal research i the field of istabilit of the o-liear mechaical sstems was doe b Todl [ ]. The mai coclusios were published ma times. The state formulas for calculatig the frequec of parametric resoaces ad describe the istabilit itervals for the frequec of parametric ecitatio if the mechaical sstem cotais ol oe elemet with the periodic chage of the parameter accordig to the siusoidal fuctio of time. ICSV eijig Chia -7 Jul

2 st Iteratioal Cogress o Soud ad Vibratio (ICSV) eijig Chia -7 Jul ow we will use the summar of a paper writte b Petermeier ad Ecker [5] who distiguish Pr betwee a Priciple Parametric Resoaces at frequecies ad Combiatio Parametric Resoaces at frequecies where j k jk j. These frequecies are defied as follows Pr j j k j jk j k () ad are the j-th ad k-th atural frequec of the liear sstem. The deomiator represets the order of the parametric resoaces. Horst Ecker is developig methods for parametric dampig of mechaical sstems ad carries out eperimets i this field dampig at the Viea Uiversit of Techolog. Ulike Petermeier ad Ecker [5] who use the catilever beam model of the Timosheko tpe this paper eamies the effect of parametric cotrol based o the lumped parameter model of the Euler-eroulli tpe as the beam is thi ad the Euler-eroulli theor is best to use for desig the model. The simulatio of the beam vibratio deca ad the radom ecitatio is a tool for aalzig the metioed effect. The mai objective of the article is to demostrate a positive effect of parametric ecitatio o dampig of the catilever vibratio with the use of a piezoactuator patch.. Mathematical model of catilever beams The mathematical model should be simple eough that it could be created i the Matlab- Simulik eviromet as a lumped parameter model. The resoat frequec ad deflectio of the beam of the discrete elemets should match the beam as a cotiuum. It is assumed that the beam is combied from rigid elemets which are coected b fleible liks formed b torsio sprigs as it is show i Fig.. It is also assumed that the bedig stiffess K of the fleible liks of the adjacet elemetar beams relates the applied bedig momet to the resultig relative rotatio of the elemetar beams ad the correspodig potetial eerg. ll the bedig stiffess will be idetical ecept for the oe coectio whose stiffess would be periodicall chaged. The periodic chages i the bedig stiffess ca be achieved b usig the patch piezoactuator which is glued to the surface of the beam ad is ot coected with the rigid frame. The effect of this pheomeo will be discussed at the ed of this chapter. This assumptio will chage the sstem o oliear ad o-statioar. Figure. Coordiates of elemets of a catilever beam. The multibod sstem i Fig. is associated with the Cartesias coordiates z. The catilever beam is clamped at the -plae ad its ceterlie is parallel to the z-ais. It is assumed ol a plaar motio of the catilever beam i the z-plae. The lik of a pair of the adjacet beam elemets is cosidered i the metioed plae as free with the metioed torsio sprig. To avoid the additioal set of costrais for the lik of the idividual beam elemets i oe poit the coordiate sstem is chose i such a wa that describes motio of the meetig poits of two adjacet eleme- ICSV eijig Chia -7 Jul

3 st Iteratioal Cogress o Soud ad Vibratio (ICSV) eijig Chia -7 Jul tar beams called odes [6]. The vertical coordiates of these odes are desigated b. The agle of rotatio of the elemetar beams with respect to the horizotal ais ca be desigated b if all agles ad their measure i radias ca be epressed b it is valid are small eough. For ad therefore it is assumed that. The coordiates of the beam equidistat poits i the Cartesia coordiates ad the idepedet geeralized coordiates for agrage's equatios of motio are idetical. For further derivatio it makes sese ol the motio i the directio of the -ais. ecause the are assumed small deformatios the shifts of the odes i the directio of the z-ais are eglected. The potetial eerg V of the catilever beam is as follows V K. K () For the beam i the horizotal positio the force of gravit is simpl added to the force actig at the elemet. The kietic eerg T of the catilever beam as a cotiuum which is replaced b its lumped parameter model is as follows T dy m dt J d dt where J is the momet of iertia [kg m ] about the ais which is situated the horizotal ad it is perpedicular to the ceterlie of the elemetar beam. The catilever beam is a coservative sstem. agrage's equatios of motio of such a sstem are as follows d T dt T V.... The first derivatives of the potetial eerg V i the agrage's equatios with respect to the variables k k... ad k k... were used to create the stiffess matri. The ostatioar stiffess K depeds either o k or o k which is missig i the formula (5) cos. K K t (5) The ol problem is that three equatios of motio cotai the stiffess K as show i the followig formulas [6]. fter the substitutio from (5) it ca be obtaied where : : :... K... K... K K... K t K... K t K... K t t cos t cos. (7) t The derivatio of the equatios of motio has bee published previousl so it will iclude ol the most importat formulas [7 8] with the use of agrage's equatio. fter itroductio smbols M for a mass square matri K for a stiffess square matri ad T for a coordiate colum vector ito the matri equatio of motio we obtai the equatio for forced vibratio M K t (8) F e () () (6) ICSV eijig Chia -7 Jul

4 st Iteratioal Cogress o Soud ad Vibratio (ICSV) eijig Chia -7 Jul ICSV eijig Chia -7 Jul where. cos T e t K F (9) It is supposed that the cross sectio of the beam is a rectagular. The momet of iertia of the beam elemet about the horizotal -ais ad perpedicular to the ceterlie of the beam is calculated accordig to the formula h m J where h is height ad m is mass of the elemet. The agrage's equatio gives the mass matri M of the followig form. h m h m M () Similarl as for the mass matri M the stiffess matri K is created with the use of the agrage's equatios. ssumig that the bedig stiffess of all the elemet joit of is the same the we ca write K K () Hadbooks of mechaics [9] state formulas for the deflectio of the catilever beam of the legth at the free ed. The beam is cosidered as a cotiuum. This deflectio due to the force F actig at the free ed i the directio of the z-ais is as follows EI F () where m. E is Youg s modulus of the beam material bh I is the area momet of iertia of the beam cross-sectio about the horizotal -ais. The statioar bedig stiffess of each fleible lik i Fig. is desigated as K.... The statioar values of all the bedig stiffess is the same ad equal to EI K () where is a multiplicatio factor which will be discussed i the et paragraph sectio. The value of the multiplicative factor i equatio () was desiged so that the beam deflectio at the free ed of the lumped-parameter model fits the deflectio which was calculated usig the formula F K ()

5 st Iteratioal Cogress o Soud ad Vibratio (ICSV) eijig Chia -7 Jul where F F T is a force actig at the ceter of gravit. It was foud out that the value of this factor depeds o the umber of elemetar beams. The results are show i Table. Icreasig the umber of elemets meas that the factor teds to oe which meas that the bedig stiffess teds to K EI ad the discrete catilever beam teds to a cotiuous beam. Selectio of = 5 will be used i the simulatios ad therefore the correctio factor plas a importat role. The positive cosequece of the itroductio of the correctio factor is also that the resoat frequecies of a beam calculated for the beam as a cotiuum coicide with frequecies which are calculated for the beam which is divided i the elemets with the use of the eigevalues of the matri K M. Table. Multiplicatio factor for calculatio of bedig stiffess. umber 5 Factor.875. umber 5 Factor Figure. Matlab-Simulik TI model of the catilever beam for arbitrar umber of elemets. s metioed earlier the part of the stiffess of the -th sprig i Fig. is a siusoidal fuctio of time. The agular frequec of the parametric ecitatio is give b the formulas (). The parameter which represets the amplitude of chages i stiffess is searched eperimetall with the use of simulatio.. The presece of viscous dampig such as a dissipative force eteds the left side of the equatio of motio b a additioal term which is proportioal to velocit t C M K M C K F (5) e where the matri of proportioalit C for Raleigh dampig is a liear combiatio of the mass ad stiffess matrices. The relatioship to the dampig ratio ca be see usig the formula f f where f is the frequec i hertz [] where the costat of proportioalit are. 59 s ad. s as follows. Matlab-Simulik model. The equatio of motio () is the secod order ordiar differetial equatio. fter the itroductio of this substitutio ad. the the secod order equatio of motio is divided ito two ordiar differetial equatios of the first order M F M e C M where M K M C or M are parameters i the form of matrices. arragemet of the sub- TI sstem which models the catilever beam for a arbitrar umber of elemets i the Matlab- Simulik eviromet is show i Fig.. Eterig the simulatio is complete with the iitial codi-. The blocks of the Gai tpe cotai matri ad their iput is a tios ad K (6) ICSV eijig Chia -7 Jul 5

6 st Iteratioal Cogress o Soud ad Vibratio (ICSV) eijig Chia -7 Jul vector ad therefore the output is a vector as well. The simulatio model (6) of the parametric ecitatio of the beam which is divided ito 5 elemets ad the patch piezoactuator is betwee the third ad fourth elemet as is show i Fig. ad 5. The effect of active vibratio cotrol is ofte demostrates o the vibratio deca of the beam which is bet ito a statioar deflected positio b actig the force of ad the is suddel released. The iitial coditios are as follows. (7) The simulatio cocers the catilever beam of the followig parameters: =.5 m b =. m h =.5 m. The beam is divided ito 5 elemets ad therefore has 5 resoat frequecies. The value of these frequecies i Hz ad i radias per secod are listed i Table. Table. Resoat frequecies. Ide 5 f [Hz] k k [rad/s] Figure. equivalet arragemet to the parametric ecitatio. Figure 5. Matlab-Simulik model of the beam for five elemets ad feedbacks.. Simulatio of vibratio deca The effect of parametric ecitatio will be evaluated o the base of the deca rate from the iitial beam deformatio. The deca of vibratio at the free ed of the catilever beam with ecitatio switch-off is show i Fig. 6. The deca of vibratio is show o the upper pael ) i Fig. 6. The lower pael ) shows the deca i d. Eample of the effect of the parametric ecitatio o the deca rate for the frequec which is the Combiatio Parametric Resoace of the first order is show i Fig. 7. The half the amplitude of ecitatio is.. Pr The effect of the parametric ecitatio o the deca rate for the frequec of the first ad secod order of the Priciple Parametric Resoaces was tested as well but without positive results. The parametric ecitatio at the frequecies ad does ot also improve the dampig. ICSV eijig Chia -7 Jul 6

7 st Iteratioal Cogress o Soud ad Vibratio (ICSV) eijig Chia -7 Jul Figure 6. Free vibratio deca of the free ed Figure 7. Vibratio deca of the free ed for 5. Simulatio of respose to radom force The vibratio deca from the deflected positio assesses the efficiec of dampig i the time domai. The dampig effect caot be observed i the frequec domai. We assume that a force with a frequec spectrum that is close to white oise acts at the free ed of the beam. The frequec spectrum of the deflectio free ed of the beam is calculated for the idetical cofiguratio of the feedbacks. ll the above metioed spectra are show i the paels of Fig. 8. The parametric ecitatio frequec is set at the resoat frequec i.e Hz ad half the amplitude of ecitatio is.. The best ecitatio amplitude was determied b simulatio calculatio which starts with zero amplitude. Frequec spectrum of decaig vibratio from pael ) of Fig. 6 ad 7 are show i Fig. 8. Icreasig of the deca rate is due to icreasig the d badwidth of the spectrum domiat peak. 6. Coclusios Figure 8. Frequec spectrum of the sigals i the pael ) of Fig. 6 ad 7. The parametric ecitatio is oe of the tools to icrease the efficiec of vibratio dampig. The paper eamied the Priciple ad Combiatio Parametric Resoace frequecies ad their effect o the deca rate for the catilever beam. It was foud that the greatest effect o vibratio dampig has the differece frequec betwee the first ad secod resoace frequec of the catilever beam. This frequec differece is a Combiatio Parametric Resoace frequec of the ICSV eijig Chia -7 Jul 7

8 st Iteratioal Cogress o Soud ad Vibratio (ICSV) eijig Chia -7 Jul first order. Other parametric resoace frequecies are without effect o the vibratio dampig. The frequec spectra clearl eplai wh icreases dampig of the parametricall damped sstems. The domiatig peak i the spectrum splits ito two adjacet peaks ad their magitude is reduced. The objective of this paper was to demostrate that parametric dampig reduces vibratio. The amplitude of the chages i stiffess was chose with the use of simulatio. REFERECES Krek S. ad Hoegsberg J. Equal modal dampig desig for a famil of resoat vibratio cotrol formats. Joural of Vibratio ad Cotrol 9(9) pp Todl. Metoda k určeí estabilit quasi-harmoických kmitů. (The method for the determiatio of istabilit itervals of quasi-harmoic vibratio sstems) plikace matematik 959 o. pp (i Czech) Todl. To the iteractio of differet tpes of oscillatio Proceedig of Semiar Iteractio ad Feed ack 97 Istitute of thermomechaics Czech cadem of Sciece Praque 997 pp. -8. Todl. To the problem of quechig self-ecited vibratios. cta Techica ČSV Petermeier. ad Ecker H. Vibratio suppressio of a catilever beam b ope loop cotrol of a attached stiffess elemet. Proceedigs of EOC 8 Sait Petersburg Russia Jue Jul 8. 6 Tůma J. ad Škutová J. Simulatio of ctive Vibratio Cotrol of the Catilever eam Proceedig of th Iteratioal Carpathia Cotrol Coferece (ICCC ) 8- Ma Podbáské Slovak Republic. 7 Tůma J. ad Šuráek P. Stabilit of the ctive Vibratio Cotrol of Catilever eams. Proceedigs of the th Iteratioal Coferece o Vibratio Problems (ICOVP-) isbo Portugal 9- September MPTC IS abstract p. 95 article pp.. 8 Šuráek P. ad Tůma J. Eperimets with the ctive Vibratio Cotrol of s Catilever eam. Proceedigs of the th Iteratioal Coferece o Vibratio Problems (ICOVP- ) isbo Portugal 9- September MPTC IS abstract p. 9 article pp Flugge W. (editor) Hadbook of Egieerig Mechaics McGrow Hill 96. Hi J. ad Fu Z-F. Modal alsis utterworth Heiema Oford. ckowledgmet This research has bee supported b the Czech Grat gec project o. P//5 ctive vibratio dampig of rotor with the use of parametric ecitatio of joural bearigs ad elaborated i the framework of the project Opportuit for oug researchers reg. o. CZ..7/../.6 supported b Operatioal Programme Educatio for Competitiveess ad co-fiaced b the Europea Social Fud ad the state budget of the Czech Republic. ICSV eijig Chia -7 Jul 8