BOUNDARY ELEMENT METHODS FOR VIBRATION PROBLEMS. Ashok D. Belegundu Professor of Mechanical Engineering Penn State University

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1 BOUNDARY ELEMENT METHODS FOR VIBRATION PROBLEMS by Aho D. Belegundu Profeor of Mechancal Engneerng Penn State Unverty ASEE Fello, Summer 3 Colleague at NASA Goddard: Danel S. Kaufman Code 5

2 MY RESEARCH IN LAST FEW YEARS Noe reducton pave approache V m Helmholtz Reonator Noy Structure TVA/BBVA Stffener Dynamc analy Acoutc analy

3 MOST RECENT RESEARCH Parallel optmzaton algorthm Speedup number of proceor bracetng nterval reducton total lne earch lnear peedup reference 3

4 OBJECTIVES AT NASA: SUMMER 3 Tutoral ON Boundary Element Analy BEM Computer code for tutoral and problem olvng Study the potental for BEM n Vbraton at mdfrequency level.e. fll the gap beteen FEA and SEA n ve of: Lo freq FEA very good Hgh freq -- SEA eem adequate although detaled repone not obtanable Ae the tate of the art n BEM for vbraton analy

5 BEM -- SIMPLE -D EXAMPLE d d u, u, u d d u d Integrate by part tce: d u' u ' u d d d Chooe uch that d d δ Fundamental Soluton : tep 5

6 SIMPLE EXAMPLE cont d Then u u' u ' d boundary term doman term ource, and depend on ource pont Chooe ource ε. Thu and reman zero a ε. -, ', body force term -/3, and e get u' / 3 We can alo recover the oluton for all nteror pont from Eq. : u 3 /6 Note: the dfferental equlbrum eactly atfed thn the doman n BEM hle boundary condton are appromately atfed, here, the boundary only o pont, o BEM gve eact oluton 6

7 AXIAL VIBRATION & u c u Harmonc loadng: p, t u u p, t / c ω, c E / ρ c L u u p / c d Integrate by part tce, u L L L u d u p/ c Fundamental oluton : δ, n tep, u u L L L u p / c L d d 7

8 Eample u F ρ A du/d L F/c F c From Eq., tp dplacement, u tan L L Th eact, nce boundary condton are eactly atfed. Reonance occur hen co L, or L π/, 3π/,... Eq. then gve F u c Tmoheno formula: u L { tan L co L n L } F c L π L,3,5,... Conder 3 mode n the epanon FEA need about 3 element for each half ne ave - red color n fg: 8

9 Mode blue color 9

10 BEAM VIBRATION v v p ρac δ, tep, / v... { nh [ ] n [ ]} 3 But no, alo need to dfferentate v rt to get addtonal equaton for oluton

11 STATIC PLATES BY DIRECT BEM D p Krchoff theory for thn plate term corner S A ds M M V V da θ θ Chooe the concentrated force fundamental oluton uch that r r D ln hch gve, π δ Thu K F F ds M M V V da p D c S A θ θ here c equal.5 f on mooth S, and equal f nteror.

12 A econd equaton obtanable a: c ξ ξ c η η D A p ' da ' V V ' θ ' M M ' θ K ' F F' S ds here the concentrated moment fundamental oluton ' gven by ' π D r ln r co α Epreon for V, V etc are gven n the lterature. Note: Corner eg. rectangular plate requre pecal attenton Dtrbuted load requre doman ntegraton Here, a functon defned uch that, and then convertng to a r r r contour ntegral a p da p co β ds r A here β angle beteen r and n. Smlarly for '. S

13 BEM WITH CONSTANT ELEMENTS : CONTOUR INTEGRATION WS W j contant for e element n j r Gau pont ource Thu, V ds e gau quadrature j e V ds hch evaluated ung Lnear element ll provde more accuracy Fnally: A b A quare, unymmetrc, dene, dm N3K here N no. of regular node, K no of corner node Clamped Plate: [hear, moment] per unt length Smply-Supported Plate: [lope, hear] 3

14 DYNAMIC ANALYSIS OF PLATES D p D h & & ρ For general tranent loadng, apply Laplace tranform and then ue BEM. Here, e aume harmonc loadng. Thu, ampltude e t t,, ω c D p ω, Chooe the concentrated force fundamental oluton uch that [ ] 8, r H r H δ The concentrated moment fundamental oluton gven by ' r co ' r K c Y c r J c α

15 CORNER EFFECTS When ource pont a corner, to concentrated moment fundamental oluton, and correpondng equaton, need to be rtten. There are a total of three unnon at each corner When a plate loaded, corner tend to curl up corner reacton eep thee clamped Detal are omtted here for brevty 5

16 HANDLING DOMAIN PRESSURE LOADS IN DYNAMICS D A p da, D A p ' da Approach : Create a doman meh, and ntegrate careful meh degn, and ntegraton are neceary for accuracy Approach : Epre p ' m, here a are determned through regreon, and each of the non functon ψ can be rtten a. Then, the Dvergence theorem allo u to convert the doman ntegral to a contour ntegral. a ψ 6

17 DIFFICULTIES IN BEM. Fundamental oluton are hard to derve for complcated dfferental equaton. Doman ntegraton requre pecal care 3. Numercal ntegraton nvolvng beel functon need pecal care 7

18 BEM WITH MULTIPLE RECIPROCITY BEM-MRM D p Chooe the concentrated force fundamental oluton uch that, of Intead δ Chooe a the tatc fundamental oluton : r r D ln hch gve, π δ We then have gnorng corner term: S A ds M M V V da p D c θ θ A da Let etc 3,, and repeatedly ue, th, S A ds M M V V da θ θ 8

19 BEM-MRM Cont d to obtan an epreon for the dplacement K F F ds M M V V da p D c S A ˆ ˆ ˆ ˆ ˆ ˆ ˆ θ θ here p p V V V ˆ, ˆ etc, along th a mlar equaton for for the lope. 9

20 Advantage of BEM_MRM Smple fundamental oluton regardle of complety of dfferental equaton All doman ntegral can be ealy converted to contour ntegral Clamed that number of term, p, are mall for convergence th true for all frequence? Integral ndependent of can be done once and tored thn the frequency loop

21 COMPUTER CODE Crcular plate done, Rectangular plate n progre Concentrated load, Contant boundary element Dampng: E E η Input: preure a SPL converted to pont load Output: Center dplacement V frequency PSD b N b n [ acc n ], here f and u acc accn N b fu fl, fl upper & loer freq n band, accelerat on n g', GRMS - ω / g, ω f / π n n n n [ acc n ] n at Hz nc. Valdaton: th Tmoheno formula and th Any

22 VALIDATION EXAMPLE Clamped Crcular Steel Plate, Rm, η.3 Concentrated load N.5-5 Crcular Plate Center Dplacement frequency, Hz

23 ANSYS RESULT Crcular plate concentrated 3

24 SAMPLE PROBLEM Clamped Crcular Steel Plate m radu, cm thc, η. Preure correpond to about 5 Pa /3-octave band, rangng from 6Hz to Hz 5 SPL n /3 Octave Band SPL, db

25 CENTER DISPLACEMENT V HZ 6-3 Crcular Plate 5 Center Dplacement frequency, Hz 5

26 MEAN ACCELERATION IN G S IN EACH /3 RD OCTAVE BAND 8 Crcular Plate 7 Center Acceleraton n g"

27 PSD IN EACH /3 RD OCTAVE BAND Crcular Plate.8.6 g /Hz ntgf /deltaf/deltaf

28 DRAWBACK IN MASS MATRIX CALCULATIONS IN FINITE ELEMENTS u N q u dplacement feld n element e q nodal dplacement of e N hape functon polynomal Ma matr: T m N N dv e M aembled from each m Choce of N crtcal n dynamc, nce nertal force equal ω M, and oluton obtaned from [K - ω M] - F Comment: for large ω, polynomal hape functon are nadequate Alo, hear effect mut be properly captured 8

29 A FEW KEY REFERENCES Boo: Boundary element an ntroductory coure, nd ed, C.A. Brebba and J. Domnguez Paper: Boundary ntegral equaton for bendng of thn plate, Morr Stern Paper: free and forced vbraton of plate by boundary element, C.P. provda and D.E. Beo, Computer method n appled mechanc and engneerng, 7, 989, 3-5 Paper on BEM-MRM: ee J. Slade. Eample: Multple recprocty method for harmonc vbraton of thn elatc plate, V. Slade, J. Slade, M. Tanaa, Appl. Math. Modellng, 7, 993,

30 SUMMARY OF WORK COMPLETED MAY 9 TH JULY 5 TH, 3 Tutoral ha been developed for BEM n vbraton -- rod, beam, Laplace eq., plate bendng Computer code for crcular plate vbraton clamped or mply-upported valdated th Cloed-form oluton and Any SPL nput, PSD and and other metrc output State of the art for vbraton analy ha been aeed Rectangular and other geometre on-gong Reearch ue dentfed ee net lde 3

31 FUTURE WORK IN APPLYING BEM FOR PLATE VIBRATIONS at MID-FREQUENCIES Accurate and effcent numercal ntegraton of body force term Develop BEM-MRM and compare th tandard BEM; more valdaton Include hear effect BEM-MRM attractve Generalze geometre any planar hape, thc plate, hell Compare BEM, FEM, Eperment on a pecfc problem; Spectral element method Other reearch: hoc loadng, optmzaton 3

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