Added mass estimation of open flat membranes vibrating in still air

Transcription

1 Added mass estimatio of ope flat membraes vibratig i still air *Yua-qi Li 1),Yi Zhou ), Akihito YOHIDA 3) ad Yukio TAMURA 4) 1), ) Departmet of Buildig Egieerig, Togji Uiversity, haghai, 9, Chia 1) liyq@togji.edu.c 3), 4) Wid Egieerig Research Ceter, Tokyo Polytechic Uiversity, Kaagawa, 43-97, Japa ABTRACT I fluid mechaics, the added mass or virtual mass, is the added iertia to the system, sice the icrease or decrease i the body acceleratio should cause the fluid to move aroud the body i such a way that the object ca move through it, ad the body ad the fluid caot simultaeously occupy this physical space. For light weight structures, such as membrae structures, whe they vibrate i a certai kid of fluids, a part of the surroudig fluid will be ivoked ad will vibrate together with structures. Hece, the added mass should have a sigificat ifluece o the vibratio of membrae structures. However, the research o the added mass of flexible structures is still limited. I this study, a framework to umerically aalysis the added mass of ope flat membraes has bee established by usig the Boudary Elemet Method (BEM). Two added mass ls are discussed, oe oly cosiderig the effect of the membrae geometric shape, ad the other cosiderig the effect of both the geometric shape ad shape of membraes. To compare the two added-mass l, a vacuum chamber was desiged to test the vibratio of a circular flat membrae i still air with various air pressures. The results showed that the estimatio of the added mass by the proposed approaches based o the boudary elemet method was reasoable ad suitability 1. ITRODUCTIO I fluid mechaics, the added mass or virtual mass, is the added iertia to the system, sice the icrease or decrease i the body acceleratio should cause the fluid to move aroud the body i such a way that the object ca move through it, ad the body ad the fluid caot simultaeously occupy this physical space. For light weight structures, such as membrae structures, whe they vibrate i a certai kid of fluids, a part of the surroudig fluid will be ivoked ad will vibrate together with structures. Hece, the added mass should have a sigificat ifluece o the vibratio of membrae structures. 1),3),4) Professor ) Doctor tudet

2 Actually, the added mass of typical objects, such as cyliders ad spheres, movig i fluid with acceleratio had bee widely ivestigated. However, the research o the added mass of flexible structures is still limited. Up to ow, several umerical simulatio methods had bee developed to ivestigate the added mass of flexible structures. Irwi ad Wardlaw (1979) preseted a empirical equatio for estimatig the added mass for the membrae roof of Motreal tadium. With the framework of the thi airfoil theory, Miami (1998) had ivestigated a membrae with its eds fixed i a icompressible fluid, ad it was proposed that the added mass is equivalet to the air uiformly distributed o the membrae with a estimated height of 68% the legth of the membrae. Yadiki et al. (3) reviewed the fluid loadig formulatios ad applied the thi airfoil theory to umerical study of the fudametal properties of the added mass of a flexible plate oscillatig i fluid. ygulski (1994) preseted a method for solvig the free vibratio ad the liear forced harmoic vibratio problems for ope membrae structures iteractig with air, ad the method describig the aerodyamic pressure was based o the boudary itegral equatio, which was solved by the Boudary Elemet Method (BEM). ygulski (1997) aalyzed the problem of iteractio betwee a peumatic structure ad the surroudig air by usig the BEM ad FEM. However, o test result had bee applied to verify the results by the umerical simulatio methods metioed above. ewall, et al (1983) udertook a experimetal ivestigatio of membrae vibratios. Tests were preformed both i air ad i vacuum for various membrae pretesios. ewall, et al. (1983) also proposed a distributio l of the added mass of the membrae. I this study, a framework to umerically aalysis the added mass of ope flat membraes has bee established by usig the BEM. The velocity potetial of the still air satisfies the Laplace equatio, ad the boudary coditios o the surface are of the euma type. The aerodyamic pressure is described by the boudary itegral equatio, ad solved by the BEM. Two added mass ls are discussed, oe oly cosiderig the effect of the membrae geometric shape, ad the other cosiderig the effect of both the geometric shape ad shape of membraes. I order to verify umerically aalysis method, a vacuum chamber was desiged to test the vibratio of a circular flat membrae. The results showed that the estimatio of the added mass by the proposed approaches based o the boudary elemet method was reasoable ad suitability.. The added mass of ope flat membraes.1 umerical aalysis of still air iduced by ope flat membrae A light ope membrae structure of ay shape i still air is cosidered. The membrae durig vibratio will iduce the motio of surroudig air, ad the air becomes a source of the additioal iertia forces, as the same as the cotributio of the structural mass. It is assumed that the air is icompressible ad iviscid, ad the velocity potetial of the air satisfies the Laplace equatio, i.e., φ φ φ (1) x y z where φ is the velocity potetial of the air.

3 The solutio of this equitatio i itegral form is φp 1 4 Q ( ) d φ i r P P Q PQ where, φ Q is the velocity potetial of the air at poit Q o the surface, φ P is the velocity potetial i ay poit of the space, r PQ is the distace betwee ay poit P ad a poit φp Q o the surface (as Fig.1 show), ad is the air velocity ormal to the surface at P the poit P. () Fig.1 A light ope membrae structure The boudary coditio o the surface is of euma's type ad it is a couplig coditio betwee the structure ad the air. The formulatios of the aerodyamic pressure ad acceleratio of the air are φ φ p, a (3) t t where is the air desity. Differetiatig Eq. () with respect to time ad usig to Eq. (3) yields 1 4 ap pq ( ) d (4) i P Q rpq where p Q is the resultat aerodyamic pressure actig at the poit Q. The BEM is used to umerically solve the boudary itegral equatio, Eq. (4). The surface of the membrae structure is discretized usig the triagular elemets. The boudary elemet discretizatio of Eq. (4) results i the followig 4 a Ap (5) where, the matrix A, a complex matrix ( is the umber of triagular elemets) is

4 A i 1 ( ) d r P Q PQ The kerel of the itegral has a strog sigularity of the r -3 order, whe the poit Q approaches the poit P (r PQ ). For this case the itegral caot be directly determied. The hypersigular itegral (6) is evaluated aalytically ad umerically usig the Gree's theorem chagig the surface itegral ito the cotour itegral (ygulski, 1994). The resultat aerodyamic forces P actig at the cetre of the ith elemet ca be calculated as Pi pii, where i is the area of the ith elemet. The relatioship betwee the aerodyamic forces P actig o the odes ad the resultat aerodyamic forces P actig at the cetre of the elemets ca be determied by 4 P P (7) rj ji i ri where, P rj is the rth compoet of force at ode j, ji is the jth iterpolatio fuctio at poit i, ri is the rth compoet of the uit ormal vector i. For the total system P TP (8) where T deotes the trasformatio matrix. The relatioship betwee the acceleratio of odes a i the global coordiate system ad the acceleratio at the cetre of the elemets a is as follows: (6) a T T a (9). Determiatio of added mass accordig to effect of the geometric shape Usig the Eq. (5), (8) ad (9), we ca get where the added mass matrix P M a a (1) -1 T Ma 4 TA T. The structural dyamic matrix equatio ca be obtaied by meas of FEM. The discretized equatio of vibratios of the structure accoutig for the aerodyamic forces is give by Ku CuMu P (11) s s s where K s ad M s are the stiffess ad mass metrics of the structure. The equatio of motio of a udamped dyamic system i matrix otatio is Ku ( M M) u (1) s s a I this method, the added mass is determied oly by the membrae geometric shape, ot the shape.

5 .3 Determiatio of added mass accordig to effect of the geometric shape ad the shape Ay displacemet vector u dyamic for the structure ca be developed by superposig suitable amplitudes of the ormal s (Clough ad Pezie, 3). For ay modal compoet u k, the displacemets are give by the product of the ( k ) shape vector ψ ad the modal amplitude ( k ) ; thus the structural vibratio displacemet ca be expressed as K K u ( k) ( k) u k ψ (13) k 1 k 1 The structural vibratio velocity ad acceleratio ca be expressed as K ( k) ( k) v ψ K ( k) ( k), a ψ (14) k 1 Ad the resultat aerodyamic forces actig o the structure ca be expressed as K ( k) ( k) p p (15) k 1 ( k) -1 ( k) where p = 4 A ψ. I the case of a icompressible iviscid statioary fluid, the chage rate i time of kietic eergy of ay part of the fluid is equal to the work doe by the pressures o its surface. E v i pidi t (16) where, E is the kietic eergy, ad v deotes the velocity of the fluid particle i the directio of the ormal. Assumig the fluid-solid iterface of the membrae ca be expressed as E h i p idi t (17) t where, h / t is the velocity of the membrae i the ormal directio, h is the surface displacemet of the membrae structure. The chage rate i time of kietic eergy of the flow field iduced by the kth atural vibratio ca be expressed as ( k ) E ( k) ( k) ( k) ( k) ( i i) pi di t (18) O the other had, the chage rate i time of kietic eergy of the oscillatig ope membrae ca be take as the cotributio of a equivalet mass per uit area, or the added mass, M a, as described by E hi h i M a d i t (19) t t For the kth atural vibratio, Eq. (19) ca be give ( k ) E ( k) ( k) ( k) ( k) ( k) M a ( i i)( i i) d i t () k 1

6 o a expressio for the added mass per uit area iduced by the kth atural vibratio ca be give i the followig form: ( k) ( k) ( i i) pi di ( k ) M a (1) ( k) ( k) ( )( ) d i i i i i I this method, the added mass is determied by the membrae geometric shape, as well as the shape. 3 Testig 3.1 Test set-up To compare the two added-mass l, a vacuum chamber was desiged to test the vibratio of a circular flat membrae i still air with various air pressures. The membrae was clipped by a top circle ad a bottom circle. The prestress was imposed uiformly by liftig a ier circle, as show i Fig.. Due to the limitatio of the vacuum pump, a complete vacuum could ot be achieved. The vibratio of the membrae was measured i 4 levels of air pressures, i.e., 1atm,.8atm,.6atm, ad.35atm. Three laser-displacemet sesors were used to measure the vibratio displacemet. Fig. 3 shows the arragemet of the measurig poits. Two membrae materials were used i the tests, icludig a latex ad a rubber membrae, ad five prestress levels of the membrae were tested. Table 1 shows the test cases. Top circle Membrae crew Bottom circle Liftig circle Bodig Colum LD Vacuum pump Fig. Test setup

7 Fig. 3 Arragemet of measurig poits of laser displacemet sesors (mm) Table 1 Test cases Air pressure Latex membrae Rubber membrae σ 1 σ σ 3 σ 4 σ 5 1 atm A1 B1 C1 D1 E1.8 atm A B C D E.6 atm A3 B3 C3 D3 E3.35 atm A4 B4 C4 D4 E4 *1 atm is the stadard atmosphere pressure. 3. Test results Fig. 4 shows the power spectral desity of the displacemet of the membrae vibratig i various air pressures. Apparetly, the atural frequecies of the membrae icrease as air pressure decreases, as give i Table. Compared to Fiite Elemet Method (FEM) aalysis (as give below), it seems that the 3rd vibratio of the latex membrae was lost due to the arragemet of the displacemet measurig poits, which are located at the odal lies of the 3rd vibratio. P..D Frequecy f (Hz) (a) Case A1 P..D Frequecy f (Hz) (b) Case A

8 .1. P..D P..D Frequecy f (Hz) (c) Case A Frequecy f (Hz) (d) Case A4 Fig. 4 Power spectral desity of displacemet of the circular membrae vibratig i various air pressures Table atural frequecy of the circular membrae vibratig i various air pressures Case 1st d 3rd 4th Case 1st d 3rd 4th f 1 (Hz) f (Hz) f 3 (Hz) f 4 (Hz) f 1 (Hz) f (Hz) f 3 (Hz) f 4 (Hz) A C A C A D A D B D B D B E B E C E C E Aalysis of the added mass Because added mass has sigificat ifluece o the atural frequecy of the membrae, the total added mass could be estimated from the relatioship betwee air pressure ad the atural frequecies f t of the membrae. Accordig to structural dyamics, the fudametal frequecy f s of the membrae vibratig i air ad the fudametal frequecy f t i vacuum have the followig relatioship: m / m f / f 1 () a s s t where m s is the membrae mass per uit area. Accordig to the equatio of the state of a ideal gas, the air desity ρ has the followig relatioship with the air pressure:

9 PM / RT 1.C a (3) where P is the air pressure; V is the volume of the chamber; R is the ideal gas costat; 8.31Pa.mol -1.K -1 ; T is the temperature, take as a costat value, 93K; M is molar mass of air; ad C a is the ratio of the air pressure i the chamber to the stadard atmosphere pressure. The, the fudametal frequecy of the membrae i vacuum ca be obtaied by fittig the curve of the relatioship betwee 1/f t ad ρ with the least square method, as show i Table 3. The fittig curves are preseted i Fig. 5. It ca be see from the figure that 1/f t ad ρ basically have a liear relatioship. 1/f t (s ).5 x Fittig curve 1st Fittig curve d Fittig curve 3rd 1/f t (s ) 5 x Fittig curve 1st Fittig curve d Fittig curve 3rd a (kg/m 3 ) a (kg/m 3 ) (a).9mpa (b).178 MPa 1/f t (s ) Fittig curve 1st Fittig curve d Fittig curve 3rd 1/f t (s ) 1 x Fittig curve 1st Fittig curve d Fittig curve 3rd a (kg/m 3 ) a (kg/m 3 ) (c).97mpa (d).471mpa

10 1/f t (s ) 4.5 x Fittig curve 1st Fittig curve d a (kg/m 3 ) (e).88mpa Fig. 5 Liear fits to 1/ft as a fuctio of air desity for the lowest three s of vibratio for five prestress values Table 3 atural frequecies of the circular membrae vibratig i vacuum Fiite Elemet Aalysis Fittig result of test Prestress 1st d 3rd 4th 1st d 3rd 4th (MPa) σ 1 = σ = σ 3 = σ 4 = σ 5 = Verificatio of two added-mass ls The prestress i the membrae was estimated from the followig equatio, s /.448 F m f r (4) t1 where F is the prestress i the membrae; r is the radius of the circular membrae; ad f t1 is the fudametal frequecy of the membrae vibratig i vacuum, which was derived from the test results. The, the atural frequecies of the d vibratio ad higher were estimated with the Fiite Elemet Method (FEM), as show i Table 3. The atural frequecies of the membrae cosiderig the added masses are listed i Table 4. Comparig Table ad Table 4, it ca be see that the result of the added mass l by Eq. (1) shows better agreemet with the test results i the 1st, however, the error betwee the test results ad the results with the added mass by Eq. (1) icreases as the vibratio icreases. Geerally speakig, the results with the added mass estimated by Eq. (1) are better agreemet with the test results from 1st to 4th. It also ca be see that the error betwee the test results ad the results with the added mass l decreases as the prestress icreases.

11 Table 4 atural frequecies of the circular membrae cosiderig the added mass Model by Eq.(1) Model by Eq.(1) Case 1st d 3rd 4th 1st d 3rd 4th f 1 (Hz) f (Hz) f 3 (Hz) f 4 (Hz) f 1 (Hz) f (Hz) f 3 (Hz) f 4 (Hz) A A A A B B B B C C C C D D D D E E E E Coclusios Added mass estimatio is a key issue i wid-iduced vibratio aalysis of membrae structures. I this paper, the boudary elemet method was applied to estimate the added mass for ope flat membraes vibratig i still air. Two added mass ls were proposed ad discussed, oe oly cosiderig the effect of the membrae geometric shape, ad the other cosiderig the effect of the geometric shape ad the shape of membraes. Compariso with the data from the tests o the circular membrae, it showed that the estimatio of the added mass by the proposed approaches based o the boudary elemet method was reasoable ad suitability. The mai fidigs were: 1) Added mass of air has a sigificat ifluece o the atural frequecy of membrae structures i vibratig. ) The proposed added mass l based o the effect of the geometric shape ca have a good agreemet with the test results i low-order s, ad the error will be icrease as the order of vibratio s icreases.

12 3) The proposed added mass l based o the effect of the geometric shape ad the shape ca have a better coformity with the test results both i low-order s ad high-order s. REFERECE Irwi H.P.A.H., Wardlaw R.L. (1979), A wid tuel ivestigatio of a retractable fabric roof for the Motreal Olympic stadium, Proceedigs of the 5th Iteratioal Coferece, Colorado, UA: Cermak J. E., Miami H. (1998), Added mass of a membrae vibratig at fiite amplitude, J. Fluid truct. 1, Yadyki Y., Teetov V., Levi D. (3), The added mass of a flexible plate oscillatig i a fluid, J. Fluid truct. 17, ygulski, R. (1994), Dyamic aalysis of ope membrae structures iteractig with air, J. umer. Methods Egrg,, 37(11), ygulski, R. (1997). umerical aalysis of membrae stability i air flow, J OUD VIB, 1(3), ewall J.L., Miseretio R., Pappa R.. (1983), Vibratig studies of a lightweight three-sided membrae suitable for space applicatio, AA Techical Paper, 95.