Vibration Attenuation of Air Inflatable Rubber Dams with Variable Anchorage Width

Transcription

1 9º Cogresso Nacioal de Mecâica Experimetal Aveiro, de Out., 1 Vibratio Atteuatio of Air Iflatable Rubber Dams with Variable Achorage Width Amorim, J. 1, ; Dias Rodrigues, J. 1 INEGI, Istituto de Egeharia Mecâica e Gestão Idustrial Departameto de Egeharia Mecâica, Faculdade de Egeharia da Uiversidade do Porto ABSRAC he atteuatio of mechaical vibratios is a key factor i the desig of structures, eve more whe they are related with flexible membraes as iflatable dams, where the self-excited vibratios are assumed to result from hydraulic disturbaces that iclude usteady water flow. I this work, the strategy for the atteuatio of structural vibratios i semi-cylidrical air iflatable rubber dams proposed by Choura (1997), amely by properly varyig the iteral pressure, is exteded to a variable achorage width geometry adaptig the mode shapes parameterizatio. It is demostrated that a adequate variatio i the iteral pressure ca modify the dyamic parameters of the structure. his approach is used to actively cotrol the structure, measurig the iduced vibratio ad varyig the iteral pressure accordigly for atteuatio. he methodology followed i this work may fall uder the defiitio of active cotrol oce that the iteral pressure is chaged i "real time", as a fuctio of the self-excited vibratio. 1 - INRODUCION Iflatable Rubber Dams Iflatable dams are flexible structures attached to a foudatio ad have bee used for various purposes: water supply, power geeratio, flood cotrol or eve recreatio. he simplicity ad flexibility of the rubber dam structure ad its prove reliability are key cosideratios i its scope of applicatios. heir legths rage from a few meters to several hudred meters, but their heights are usually less tha seve meters (Hsieh 199). hey are usually iflated with air, but ca also be filled with water or a combiatio of air ad water (Mysore ad Liapis 1998). - ON HE CAUSES OF VIBRAIONS Iflatable rubbers dams are flexible structures ad due to their low stiffess they chage their geometry accordigly to the pressure distributio alog the surface. he occurrece of vibratios ca be very complex i ature oce that it's a fluid/structure iteractio. Nevertheless, we ca distiguish the followig types of vibratios (Gebhardt, M., et al. 1): vibratios of the appe, vibratio due to pressure fluctuatios ad vibratios due to uplift forces..1 - Vibratios of the appe hese vibratios arise i iflatable dams with deflectors ad ca be observed at small overflow depths ad low tailwater levels. Due to the deflector, a air cavity is formed betwee the membrae ad the appe (Fig. 1). If this cavity is ot vetilated free-surface udulatios might occur but result oly i small deformatios. Fig. 1 Vibratios of the appe.

2 Vibratio Atteuatio of Air Iflatable Rubber Dams. - Vibratios due to pressure fluctuatios hese vibratios are caused by a ustable appe separatio o the dowstream face of the rubber dam (Fig. ). he cosequet pressure fluctuatios lead to vibratios ad large deformatios. Fig. Vibratios due to pressure fluctuatios (Gebhardt, M., et al. 1)..3 - Vibratios due to uplift forces hese vibratios occur durig high overflow ad icreased dowstream water level. he presece of high gradiet pressures leads to a chage i the cross sectio to a aerodyamic shape like a wig. Due to the cotractio of the flow a egative pressure area is formed o the upstream face of the rubber body which lifts the membrae ad leads to a chage of the flow field (Fig. 3). With the icreased flow resistace the membrae is pushed dow agai. Fig. 3 Vibratios due to uplift forces. (Gebhardt, M., et al. 1). membrae body to break the appe ad avoidig log air cavities ad pressure fluctuatios or vetilatio of the air cavity to elimiate the low pressures, to ame just a few. I this work, a approach to chage the dyamic properties of the rubber dam is followed measurig the iduced vibratio ad varyig the iteral air pressure accordigly for atteuatio. his methodology is appropriate to decrease the problems of cost ad costructio of splitters or air vetilatio ad may fall uder the defiitio of active cotrol oce that the iteral pressure is chaged i "real time", as a fuctio of the self-excited vibratio. 3 - MECHANICAL MODEL Equatio of Motio he dyamics of a air-iflated cylidrical membrae dam (Fig. ) ca be described by the followig equatios (Choura 1997): w μr w μr w + w + = θ θ P t P t v w = θ (1) with the followig boudary coditios for w(θ, t) ad v (θ, t), radial ad tagetial displacemets at ay time t, respectively: ( ) ( α ) ( ) ( α ) w, t = w, t = v, t = v, t = () All of these situatios share a iterestig commo trait that is the selfexcited ature of the vibratio. he vibratio of the rubber dam feeds the excitatios, i.e. water flow fluctuatios ad/or pressure gradiets, which makes the problem very challegig. here are some solutios poited by the researchers to break the cycle excitatio/vibratio amely, splitters or breakers o the Fig. Geometry of a iflatable dam. where R is the radius of the circular rubber dam model, μ is the mass per uit legth ad α is the cetral agle of the circular membrae arc. he dyamics of

3 Amorim J., Dias Rodrigues J. the air-iflated membrae is based o the assumptios that the deflectios are small, extesio ad the effect of dampig are egligible durig deformatio ad the membrae is log, which allows a twodimesioal model Cotrolled Pressure It is assumed that the iteral pressure Pt () is composed by a omial pressure P ad a cotrolled pressure P () t : Pt = P+ P t (3) () () he mai objective of this methodology is assure a miimum of iteral pressure for sustai the equilibrium cofiguratio. Moreover, the iterest is to set a law for the variatio of P () t that atteuates the vibratio of the iflatable structure, oce that the dyamic properties deped upo the iteral pressure. he radial displacemet w(θ, t ) ca be writte separately, i space ad time, as: w θ, t = W (θ) ( t) () ( ) = 1 where W () t are the eigefuctios from the problem described i Eq. 1 with costat pressure ad () t are the modal amplitudes. Hsieh (199) show that the eigevalue problem with a costat pressure leads to the followig characteristic equatio: b a (1 cos a cosh b ) + si a sih b = where: a b λ ab (5) α = 1+ λ λ + λ α = 1 λ λ + λ μωr = P (6) ad the eigefuctios that correspod to both radial ad tagetial displacemets are: θ θ W(θ) = A si a + Hcos a α α a θ θ + sih b Hcosh b b α α α θ θ V(θ) = A cos a + Hsi a a α α θ a θ + cosh b H sih b α b α (7) where by: ad A is a costat ad H b si a + sih b a = cos a cosh b W ad V satisfy: d W λ λ W H is give + (1 + d W ) = dv W = (8) (9) with the boudary coditios defied i Eq. (). With the separatio of variables method defied i Eq. (), the differetial Eq. (1) ca be re-defied as: μ P d W d W d + = ω d = t R d W P W 1+ P (1) which allows to write the modal amplitudes, i a matrix form, as: d P Ω = dt P (11) =,,, ad where the vector [ ] 1 3 the matrix = diag ( ω 1, ω, ω3, ) Ω. 3

4 Vibratio Atteuatio of Air Iflatable Rubber Dams o cotrol the vibratios of the membrae we eed to guaratee that the system of Eq. (11) is asymptotically stable. he most commo method to aalyze the stability of such systems is the Lyapuov method. I this approach, a called Lyapuov fuctio, deoted here by E ( Τ ), is a real scalar fuctio of the vector, which has cotiuous first partial derivative ad satisfies the followig coditios: 1. E( ) > for all values of ( t) d E( ). for all values of ( t) dt With the previous coditios it ca be stated that if there exists a Lyapuov fuctio for a give system, that system is stable. I additio, if the fuctio E ( ) is strictly less tha zero, the system is asymptotically stable. I mechaical systems the sum of kietic ad potetial elastic eergy is a good cadidate for the Lyapuov fuctio. akig ito accout Eq. (11) a Lyapuov fuctio ca be defied as: E( ) = + dt dt 1 d d Ω (1) Writig the time derivative of the Lyapuov fuctio ad take ito accout Eq. (11): d E( ) d d = + Ω dt dt dt P d = P dt (13) Ω For the time rate of E( ) to be egative, i.e. the eergy of the system is decreasig ad the system is stable or asymptotically stable, oe ca select P as: d P = kp dt Ω di = kp ωi i with k> (1) dt i= 1 hus, we have a law for the cotrol pressure that is a o-liear fuctio of the modal displacemets ad velocities. he costat k must be properly tued for the desired rate of covergece to equilibrium. I practice, this approach is achieved with a Proportioal ad Derivative (PD) cotroller. - CASE SUDY.1 - Iflatable Rubber Dam Let us cosider a iflatable dam with the parameters preseted i able 1. able 1 Properties of the iflatable dam. R α θ μ P (m) (rad) (rad) (Kg/m) (N/m),5 1,5 π π / 1, he first four mode shapes are depicted below: Fig. 5 1 st Mode Shape. Fig. 6 d Mode Shape.

5 Amorim J., Dias Rodrigues J. Fig. 7 3 rd Mode Shape. pressurizatio. hese results should emphasize the available possibilities i these systems to cotrol them, amely by properly varyig the iteral air pressure. For the iflatable rubber dam cosidered i this case study the associated first four atural frequecies are ω 1 = 7,9 rad/s, ω = 181,17 rad/s, ω 3 = 9, rad/s ad ω = 393,3 rad/s. Fig. 8 th Mode Shape. Aalyzig the mode shapes represeted i Fig. 5-8 it ca be see that the first mode is ati-symmetric with oe ode, the secod is symmetric with two odes ad so o. It is iterestig to ote that i the odd mode shapes oe ca distiguish that vibratio is predomiat i horizotal directio, i cotrast with the eve mode shapes where the vibratio is predomiat i vertical directio. his is importat if oe cosider the causes of vibratio where were idetified that vibratios of the appe ad due to pressure fluctuatios iduce vibratios i horizotal directio ad vibratios due to uplift forces iduce vibratios i vertical directio. Idetify the causes of vibratio ca be helpful to determie which are the predomiat modes that are excited. As oe ca see i Eq. (6) the atural frequecies are a o-liear fuctio of the iteral air pressure. I Fig. 9 the ifluece of the iteral pressure o the atural frequecies for the first four modes is represeted ad, as expected, the atural frequecies icrease with the iteral air pressure oce that the stiffess of the membrae is icreased with the Fig. 9 Ifluece of iteral air pressure i the atural frequecies. I Fig. 1 the time respose of the ucotrolled membrae dam is preseted where oly the first mode was iitially excited with 1 ( ) =,5. he time respose of the cotrolled membrae is depicted i Fig. 11 ad the respective cotrolled air pressure is represeted i Fig. 1. Fig. 1 Ucotrolled time respose of the radial displacemet. 5

6 Vibratio Atteuatio of Air Iflatable Rubber Dams Fig. 11 Cotrolled time respose of the radial displacemet. of the ucotrolled membrae dam is preseted where the first three modes were iitially excited with 1 ( ) =,5, ( ) =, 1 ad 3 ( ) =, 5. he time respose of the cotrolled membrae is depicted i Fig. 1 ad the respective cotrolled air pressure is represeted i Fig. 15. I this secod example the presece of differet vibratio modes ca be see. Nevertheless, the vibratio is atteuated as oe ca see i Fig. 1, where the cotrolled radial displacemet follows the same tedecy of the first example where oly oe mode was cosidered. he presece of differet vibratio modes leads to a irregular cotrolled air pressure (Fig. 15) however the geeral characteristics are the same of the first example (Fig.1). Fig. 1 Cotrolled iteral air pressure. he costat k was defied as: k = ω () (1) With this costat oe ca defie the gai of the system ad tue it as desired. If faster time decays are eeded the costat ca be icreased but care must be take, amely for overshootig problems that ca put i risk the structural itegrity of the rubber dam membrae. From Fig. 11 it ca be viewed that the dampig ratio varies i time, beig higher at the begiig ad low afterwards. his is due to its depedecy with the modal amplitudes. he cotrolled air pressure represeted i Fig. 1 follows the same behavior i tur of the omial pressure (5 1 N/m) but with double the frequecy. I fact, this occurs because the variable pressure P () t is sesible to the sig of the modal amplitude i () t but also to the sig of its time derivative d i /dt, as oe ca see i Eq. (1). I Fig. 13 the time respose Fig. 13 Ucotrolled time respose of the radial displacemet. Fig. 1 Cotrolled time respose of the radial displacemet. 6

7 Amorim J., Dias Rodrigues J. accordigly for atteuatio. Moreover, the strategy followed i this work ca be applied i differet systems with variable stiffess ad should emphasize the advatages of these systems (iflatable structures but ot limited to) where the dyamic properties ca be cotrolled as desired. Fig. 15 Cotrolled iteral air pressure. 5 - DISCUSSION Aalyzig Fig. 11 ad Fig. 1 it is clear that this methodology is capable of atteuate the vibratios of the rubber dam membrae. his approach is used to actively cotrol the structure, measurig the modal amplitudes ad varyig the iteral pressure as desired. he methodology followed here may fall uder the defiitio of active cotrol oce that the iteral pressure is chaged i "real time", as a fuctio of the modal amplitudes. It is worth to metio that the dampig ratio varies i time. I fact, at the begiig the dampig is high ad low afterwards. his is due to the depedecy of the dampig ratio o the modal amplitudes. It should be oted that the displacemet teds to a limit that is differet from, i.e. the vibratio is atteuated ad ot elimiated oce that the eergy is "cotrolled" ad ot etirely dissipated. It is demostrated that a proper variatio i the iteral pressure ca chage the dyamic parameters of the structure. his methodology should elighte to the advatage of varyig the iteral pressure for vibratio atteuatio. REFERENCES Choura, S Suppressio of structural vibratios of a air-iflated membrae by its iteral pressure, Composite ad Structures, 65(5), p Hsieh, J.C Free vibratio of iflatable dams, Acta Mechaica, 85, p. 7-. Mysore, G.V. ad Liapis, S.I Dyamic aalysis of sigle-achor iflatable dams, Joural of Soud ad Vibratio, 15(), p Gebhardt, M., et al. 1. O the causes of vibratios ad the effects of coutermeasures at water-filled iflatable dams, Proc. 1st. Europea IAHR Cogress. Ediburgh (CD-Rom). 6 - CONCLUSION A strategy for vibratio atteuatio of air iflatable rubber dams was followed i this work. It was demostrated that the selfexcited vibratio ca be actively cotrolled measurig the iduced vibratio ad chage the iteral air pressure 7