Simulation of the parametric excitation of the cantilever beam vibrations

Transcription

1 Simulatio of the parametric excitatio of the catilever beam vibratios Jiří Tůma, Pavel Šuráe, Miroslav Mahdal, Mare Babiuch Facult of Mechaical Egieerig VSB Techical Uiversit of Ostrava Ostrava, Czech Republic Abstract The effect of parametric vibratio dampig is theoreticall well described ad the first experimets are published. The paper focuses o the desig ad verificatio of the sstem fuctioalit which 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 priciple of goverig the piezoactuator stiffess is based o proportioal cotrol of the actuator displacemet. The trasfer fuctio of a disturbace force to the actuator displacemet is affected b the cotroller gai. Periodic chages of the cotroller gai should have a dampig effect o vibratio of the catilever beam accordig to the simulatio i Matlab-Simuli eviromet. The paper presets a simulatio stud of this method of dampig vibratios. Kewords Parametric resoaces; vibratio dampig; piezoactuators; catilever beam; variable cotroler gai; simulatio I. ITRODUCTIO A catilever beam is a simple example of a mechaical structure that ca serve as a object for testig the active vibratio cotrol. The promisig wa to reduce vibratio of mechaical structures is parametric excitatio, which is fudametall differet from the active vibratio cotrol sstems developed for example b Kre []. Ulie active vibratio dampig, which is based o the use of liear methods for cotrol of the liear time ivariat sstems, the ew wa of dampig uses a periodic chage of a parameter, usuall sprig stiffess. Such a sstem becomes o-liear ad potetiall ustable for a iterval of the frequec of the metioed parameter variatio. A fudametal research i the field of istabilit of the o-liear mechaical sstems was doe b Todl [], [3]. Todl s mai coclusios were published ma times [4]. The state formulas for calculatig the frequec of parametric resoaces ad describe the istabilit itervals for the frequec of parametric excitatio if the mechaical sstem cotais ol oe elemet with the periodic chage of the parameter accordig to the siusoidal fuctio of time. ow we will use the summar of a paper writte b Petermeier ad Ecer [5] who distiguish betwee a Priciple Pr Parametric Resoaces at frequecies ad Combiatio j Parametric Resoaces at frequecies frequecies are defied as follows Pr j. These j j j, j, j,,,...,,,... where j ad are the j-th ad -th atural frequec of the liear sstem. The deomiator represets the order of the parametric resoaces. Horst Ecer from the Viea Uiversit of Techolog is developig methods for parametric dampig of mechaical sstems ad carries out experimets i this field dampig. Ulie the paper [5], which uses the beam model of the Timosheo tpe, this paper examies the effect of parametric cotrol based o the lumped parameter model of the Euler- Beroulli tpe. The simulatio of the beam vibratio deca ad the radom excitatio is a tool for aalzig the metioed effect. The catilever beam dimesios correspod to a laborator test bech which is prepared to experimets. The mai objective of the article is to demostrate a positive effect of parametric excitatio o vibratio dampig. II. ELECTRICALLY COTROLED STIFFESS High-frequec periodic chages i stiffess ca be made ol with piezoactuators or electrodamic exciters. The use of the piezoactuators i the positio cotrol will be cosidered i this chapter. Parametric dampig requires a periodic chage i stiffess. The wa to achieve this is show i Fig.. Fig.. Variable stiffess based o piezoactuators We assume that the differece of the piezoactuator travel is proportioal to the differece of the iput voltage

2 V, where is a coefficiet of proportioalit. A effect of the differece of the exteral force F is a differece of the displacemet F, where is a compliace of the piezoactuator. The liear model of the piezoactuator is as follows V F The differece of the voltage follows the differece of the total displacemet of the piezoactuator V K 0 K, C where K C is a gai of the proportioal cotroller. The stiffess of the piezoactuator which is a part of the closed loop is give b the followig formula F K iitial stiffess C K C C The multibod sstem i Fig. is associated with the Cartesias coordiates x,, z. The catilever beam is clamped at the x-plae ad its ceterlie is parallel to the z-axis. It is assumed ol a plaar motio of the catilever beam i the zplae. The li 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 li 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 elemetar 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 axis ca be desigated b,,, ad their measure i radias ca be expressed b L if all agles are small eough. For it is valid L ad therefore it is assumed that 0 0. The coordiates of the beam equidistat poits i the Cartesia coordiates ad the idepedet geeralized coordiates for Lagrage's equatios of motio are idetical. For further derivatio it maes sese ol the motio i the directio of the -axis. Because the are assumed small deformatios, the shifts of the odes i the directio of the z- axis are eglected. A piezoactuator such as P made b the PI Compa has stiffess 38 /µm, ad because it is the low voltage tpe, the the parameter is equal to about 90 µm / 00 V. Periodic variatio i stiffess is achieved b periodicall chagig the gai of the proportioal cotroller. Coectio of the piezoactuator ito the positio cotrol loop with a time variable gai is just a proposal to prove the feasibilit of this tpe of excitatio. III. MATHEMATICAL MODEL OF CATILEVER BEAMS The mathematical model should be simple eough that it could be created i the Matlab-Simuli eviromet as a lumped parameter model. The resoat frequec ad deflectio of a beam of the discrete elemets should match the beam as a cotiuum. Because it does ot cocer a beam, which was thic, the Euler-Beroulli theor is best to use for desig a model. It is assumed that the beam is combied from rigid elemets which are coected b flexible lis formed b torsio sprigs as it is show i Fig.. It is also assumed that the bedig stiffess K of the flexible lis of the adjacet elemetar beams relates the applied bedig momet to the resultig relative rotatio of the elemetar beams ad the correspodig potetial eerg. The bedig stiffess of the flexible lis will be idetical except for the oe coectio whose stiffess would be periodicall icreased ad decreased. 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. Fig. Coordiates of elemets of a catilever beam 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 Lagrage's equatio. After itroductio smbols M for a mass square matrix, K for a stiffess square matrix ad,,, T for a coordiate colum vector ito the matrix equatio of motio we obtai the equatio for free vibratio M K 0 It is supposed that the cross sectio of the beam is a rectagular. The momet of iertia of the beam elemet about the horizotal x-axis ad perpedicular to the ceterlie of the beam is calculated accordig to the formula J x ml h where h is height ad m is mass of the elemet. The Lagrage's equatio gives the mass matrix of the followig form

3 B A We have prepared a specime of the catilever beam of the followig parameters: L = 0.5 m, b = 0.04 m, h = m to M A B A, be tested. The value of the multiplicative factor i equatio (0) was desiged so that the beam deflectio at the free ed A B of the lumped-parameter model fits the deflectio, which was calculated usig the formula (9). The deflectio of the beam where alog its legth ca be calculated with the use of the formula m h m, h A B L L Similarl as for the mass matrix M the stiffess matrix K is created with the use of the Lagrage's equatios. Assumig that the bedig stiffess of all the elemet joit of is the same, the we ca write 6 K K L Hadboos of mechaics [9] state formulas for the deflectio of the catilever beam of the legth L 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-axis ad correspods to the coordiate i Fig. therefore we ma state where F L 3 3E I x E.40 m is Youg s modulus of the beam 3 material, I x bh is the area momet of iertia of the beam cross-sectio about the horizotal x-axis. The statioar bedig stiffess of each flexible li i Fig. is desigated as K,,,...,. The statioar values of all the bedig stiffess is the same ad equal to EI x K, L where is a multiplicatio factor which will be discussed i the ext paragraph sectio. TABLE I. MULTIPLICATIO FACTOR FOR CALCULATIO OF BEDIG STIFFESS umber 5 0 Factor umber Factor K F0 where T F 0 0,0,,F 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 x L 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 matrix K M. IV. EFFECT OF THE STIFFESS PERIODIC CHAGE As metioed earlier, the part of the total stiffess of the -th sprig i Fig. is a siusoidal fuctio of time 0 t, K cos K where 0 is a agular frequec of the parametric excitatio ad is a parameter which represets the amplitude of chages i stiffess. I this paper, this amplitude was searched experimetall. The potetial eerg V of the deflected beam is as follows V L K. K 0 0 The first derivatives of the potetial eerg V i the Lagrage's equatios with respect to the variables,,,... ad,,,... were used to create the stiffess matrix. The o-statioar stiffess K depeds either o or o which is missig i the formula (3). The ol problem that rigidit stiffess i the three equatios of motio as show b the followig formulas [5] : : :... K... K... K L L L

4 After the substitutio from (0) ad () it ca be obtaied where : : :... 6K L... 6K L... 6K L... K... K... K L L L t t t t cos 0t cos t. 0 Parametric excitatio ma be replaced b three forces that itroduce a periodic bedig momet whose amplitude is proportioal to the agle betwee the adjacet beam elemets. The presece of viscous dampig, such as a dissipative force, exteds the left side of the equatio of motio b a additioal term which is proportioal to velocit M C K Ft, C M K, where the matrix 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 [0], where the costat of proportioalit are as follows 0.59 Hz Hz. ad The vector of forces i the right side of the equatios of motio (5) has periodic compoets of the same frequec ad phase. The static compoet of this vector is missig. The amplitude of the siusoidal fuctios is proportioal to the coordiate of the appropriate li poit. e t, 0,,,, 0,. F K L T elemets. These forces act o the metioed elemets i the directio of the -axis. The model of the beam is supplemeted b three feedbacs with a gai which varies periodicall over time as is show i Fig. 3. V. MATLAB-SIMULIK MODEL The equatio of motio (6) is the secod order ordiar differetial equatio. After the itroductio of this substitutio x, ad x. the the secod order equatio of motio is divided ito two ordiar differetial equatios of the first order x x x M F M Cx M K x where M K, M C or M are parameters i the form of matrices. A arragemet of the subsstem which models the catilever beam for a arbitrar umber of elemets i the Matlab-Simuli eviromet is show i Fig. 4. Eterig the simulatio is complete to the iitial coditios x0 0 ad x0 0. The blocs of the Gai tpe cotai matrix ad their iput is a vector, ad therefore the output is a vector as well. Simulatio of the parametric resoace of the beam which is divided ito 5 elemets ad the first joit of variable bedig stiffess K is i the clampig poit. It is expected that this method of parametric excitatio will have the greatest effect. The Matlab-Simuli model is show i Fig. 5. Uder this assumptio, the sstem has ol oe feedbac. The model of the beam i Fig. 5 is mared as a Subsstem i Fig. 4. The first derivative of the beam elemet displacemets with respect to time is ot eeded for parametric excitatio. 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 0 ad the is suddel released. The iitial coditios are as follows 0 T K 0, 0,..., 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. Fig. 3 A equivalet arragemet of the force excitatio to the parametric excitatio The parametric excitatio with the use of the periodic chage i the bedig stiffess ca replace three periodic forces with a amplitude proportioal to the istataeous agle betwee the pair of the -th ad ( )-th beam Fig. 4 Matlab-Simuli lumped parameter model of the catilever beam for arbitrar umber of elemets TABLE II. RESOAT FREQUECIES

5 Idex f [Hz] [rad/s] A) ad B) have the same meaig as i Fig. 6. Depedece of the deca time for a decrease of 40 db o the parameter is show i Fig. 7. The most efficiet dampig is achieved for The deca time for the excitatio frequec 3 is almost costat for the iterval of as is show i Fig. 9. The effect of the idividual value of the parameter o the frequec spectrum compositio is ot the same as is show i the ext chapter. Fig. 5 Matlab-Simuli model of the beam for five elemets ad feedbacs Fig. 7 Deca time vs. parameter µ for VI. SIMULATIO OF VIBRATIO DECAY The effect of parametric excitatio 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 excitatio switch-off is show i Fig. 6. The deca of vibratio with excitatio switch-off is show o the upper pael A) i Fig. 6. The lower pael B) shows the deca i db db. 0log 5 log 5 The time iterval of the decrease i the amplitude of istataeous vibratio from the level of 0 db to -40 db ma be a measure of dampig efficiec ad ca be called as the deca time. t0 Fig. 8 Vibratio deca of the free ed for The effect of the parametric excitatio o the deca rate for the frequecies 3 3 ad 3 3 of the first order of the Combiatio parametric resoaces is show i Fig. 9. The deca time for the excitatio frequec 3 decreases up to the value of 0. while this deca time icreases for 0.. Fig. 6 Free vibratio deca of the free ed Firstl, the effect of the Combiatio Parametric Resoaces is aalzed. The effect of the parametric excitatio o the deca rate for the frequec of, i.e Hz, is show i Fig. 8. The paels

6 Fig. 9 Deca time vs. parameter µ for 3 ad 3. The effect of the parametric excitatio o the deca rate for the Priciple Parametric Resoace Frequecies is almost without a positive effect. VII. SIMULATIO OF RESPOSE TO RADOM FORCE The vibratio deca from the deflected shape of the catilever beam assesses the efficiec of dampig i the time domai. The dampig effect ca be observed i the frequec domai. It is assumed that a radom force with a frequec spectrum that is close to white oise acts at the free ed of the beam. The frequec spectrum of the deflected edpoit of the beam is calculated for the idetical cofiguratio of the feedbac. All the above metioed spectra are show i the paels of Fig. 0. As is evidet from frequec respose spectra i Fig. 0 the domiat pea of the spectrum is split i two sub-peas while this chage of the frequec spectrum is accompaied b decreasig the magitudes of the sub-peas. The parametric excitatio frequec is set at the atiresoat frequec. The gai of the feedbac was the largest possible as foud experimetall. Fig. 0 Frequec spectrum of the excitatio ad respose at the beam free ed for VIII. COCLUSIOS The parametric excitatio is oe of the tools to icrease the efficiec of vibratio dampig. The paper examied 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 first order. The optimum size of half the amplitude of excitatio for this frequec was also determied. Other parametric resoace frequecies are without effect o the vibratio dampig. The frequec spectra clearl explai wh icreases dampig of the parametricall damped sstems. The domiatig pea i the spectrum splits ito two adjacet peas ad their magitude is reduced. The objective of this paper was to demostrate that parametric dampig reduces vibratio. Amplitude chages i stiffess were chose b the simulatio approach. ACKOWLEDGMET This research has bee supported b the Czech Grat Agec project o. P0//50 Active vibratio dampig of rotor with the use of parametric excitatio of joural bearigs ad elaborated i the framewor of the project Opportuit for oug researchers, reg. o. CZ..07/.3.00/30.006, supported b Operatioal Programme Educatio for Competitiveess ad co-fiaced b the Europea Social Fud ad the state budget of the Czech Republic. REFERECES [] S. Kre, ad J. Hoegsberg, Equal modal dampig desig for a famil of resoat vibratio cotrol formats. Joural of Vibratio ad Cotrol, 9(9), 0, pp [] A. Todl, Metoda určeí estabilit quasi-harmoicých mitů. (The method for the determiatio of istabilit itervals of quasi*harmoic vibratio sstems), Apliace matemati, 4, 959, o. 4, pp (i Czech) [3] A. Todl, To the iteractio of differet tpes of oscillatio, Proceedig of Semiar Iteractio ad Feed Bac 97, Istitute of thermomechaics, Czech Academ of Sciece, Praque 997, pp. -8. [4] A. Todl, To the problem of quechig self-excited vibratios. Acta Techica ČSAV, 43, 998. [5] B. Petermeier ad H. Ecer, Vibratio suppressio of a catilever beam b ope loop cotrol of a attached stiffess elemet. Proceedigs of EOC 008, Sait Petersburg, Russia, 30 Jue, 4 Jul, 008. [6] J. Tůma, ad J. Šutová, Simulatio of Active Vibratio Cotrol of the Catilever Beam, Proceedig of 3th Iteratioal Carpathia Cotrol Coferece (ICCC), 8-3 Ma 0, Podbásé, Slova Republic, IEEE Catalog umber: CFP4L-CDR, ISB: [7] J. Tůma, ad P. Šuráe, Stabilit of the Active Vibratio Cotrol of Catilever Beams. Proceedigs of the th Iteratioal Coferece o Vibratio Problems (ICOVP-03), Lisbo, Portugal, 9- September, 03, AMPTAC, ISB , abstract p. 95, article pp.. [8] P. Šuráe ad J. Tůma, Experimets with the Active Vibratio Cotrol of s Catilever Beam. Proceedigs of the th Iteratioal Coferece o Vibratio Problems (ICOVP-03), Lisbo, Portugal, 9- September, 03, AMPTAC, ISB , abstract p. 39, article pp. 7. [9] W. Flugge (editor), Hadboo of Egieerig Mechaics, McGrow Hill, 96. [0] J. Hi, ad Z-F. Fu, Modal Aalsis, Butterworth Heiema, Oxford 00.