14 th Internatonal Users Conference Sesson: ALE-FSI Statstcal Energy Analyss for Hgh Frequency Acoustc Analyss wth Zhe Cu 1, Yun Huang 1, Mhamed Soul 2, Tayeb Zeguar 3 1 Lvermore Software Technology Corporaton 7374 Las Postas Road, Lvermore CA 94551 2 Llle Unversty Laboratore Mecanque de Llle, France 3 Jaguar Land Rover Lmted, Coventry UK Abstract The present work concerns about the new capablty of n solvng hgh frequency vbraton and acoustc problem, usng statstcal energy analyss (SEA). As the frequency ncreases, the number of modes ncreases. As the result, the tradtonal numercal methods lke fnte element (FEM) and boundary element method (BEM) are dffcult to use due to the large number elements requrement. It s more practcal to consder average responses and ther dstrbuton over the structure, usng a technque such as statstcal energy analyss (SEA). In ths paper, several numercal examples are nvestgated by SEA method wth. The numercal results are n good agreement wth other code. Introducton Statstcal Energy Analyss (SEA) s a structural-acoustc method that s wdely used for hgh frequency analyss. SEA arose durng the 1960 s n the aerospace ndustry to predct the vbratonal behavor when desgnng space craft. Durng ths tme computatonal methods were avalable but the sze of the models that could be handled and the computatonal speed were such that only few of the lowest order modes could be predcted [1]. Furthermore, n tradtonal analyss of mechancal vbraton the lowest modes are usually of most nterest because these modes normally have the greatest dsplacement response. But when desgnng and constructng large and lghtweght aerospace structures t s apparent that also hgh frequency broad-band loads s mportant n the process of predctng structural fatgue, equpment falure and nose producton. The name SEA ponts out certan aspects of the method. Statstcal accents that the system beng consdered are a member of a populaton of smlar systems wth known dstrbutons of the subsystem parameters. Energy s the varable of nterest. It descrbes the behavor of the system n terms of stored, dsspated and exchanged energes of vbraton. Other often used varables for acoustc and structural vbraton, such as dsplacement, pressure etc. can be derved from the energy of vbraton. Analyss emphaszes that SEA s a framework rather than a specfc technque. The development and use of SEA proved to be a good method to predct hgh frequency loads and the analyss technque has snce then been appled, extended and developed for a growng number of applcatons. For example t has been used to model sound and vbraton transmsson n buldngs, cars, arcrafts, shps and trans [2] and [3]. June 12-14, 2016 1-1
Sesson: ALE-FSI 14 th Internatonal Users Conference The General Procedures of SEA One of the fundamental prncples of SEA s that the average power flow between coupled groups of dynamcal modes s proportonal to the dfference I the average modal energes. Ths makes t possble to analyze the dynamcal response of a system consstng of many resonant modes n a certan frequency range by dvdng the modes nto groups (subsystems) and consderng a power balance equaton of the form Pn Pout for each mode group. The analyss s statstcal n that the expected value and varance of the power flow are evaluated assumng a statstcal dstrbuton n the resonance frequences of the subsystem modes [2]. The general procedures of SEA can be llustrated as shown n Fgure 1 below. Ths system conssts of two connected subsystems. Fgure1. A two subsystems SEA model. Energy flows n and out of a subsystem. The energy that flows out conssts of dsspaton P d and Pd, radaton and transmsson to other subsystems P and P. The energy that flows nto a subsystem conssts of external source exctatons P and P and transmssons from other subsystems P and P. The dsspated power n a subsystem s gven by equaton 1 and 2 below: P E (1) P d d d d E (2) Where d and d are the dampng loss factors and E and E are the total vbratonal energy of the modes at frequency f. The net transmtted power between subsystem and ( P and P ) are gven by equaton 3 and 4. P P E (3) E (4) 1-2 June 12-14, 2016
14 th Internatonal Users Conference Sesson: ALE-FSI SEA calculaton s based on energy flow equlbrum. The power balances for the two systems are gven as n equatons 5 and 6. P P P P (5) d P P P P (6) When combnng equatons 1-6, the power balance equaton for two subsystems can be expressed n matrx form shown n equaton 7. d P d E (7) P d E For a more generalzed case wth & number of subsystems, the balance equatons can be wrtten n a general form as equaton 8. P n E P E (8) Pn n n En Where equals the total loss factor and s the summaton of the dampng loss factor of the subsystem and the couplng loss factors representng energy transmsson from the subsystem to others as equaton 9. n (9) d 1, Example 1: Three plates connecton The frst example s three plates connected at a common lne shown n Fgure 2. The plates are all made of steel. Each of the plates has a sze of 1m x 1m. Plate 1 and Plate 3 are 2 mm thck, Plate 2 s 3 mm. Plate 1 s connected to plate 2 at an angle of 90, plate 3 s connected at an angle of 210. All Plates have constant dampng of 0.01. Plate 1 s excted by an nput power of 0.5 Watt (n bendng wave). We want to calculate the mean velocty ampltude of the bendng wave at each plate. Fgure 3 s the velocty of plate1 and plate 2. The results are compared wth other code [4]. June 12-14, 2016 1-3
Velocty (m/s) Velocty (m/s) Sesson: ALE-FSI 14 th Internatonal Users Conference Plate 2 Input Power Plate 3 Plate 1 Fgure2. Three plates connecton model 2.50E-02 2.00E-02 1.50E-02 1.00E-02 5.00E-03 7.00E-03 6.00E-03 5.00E-03 4.00E-03 3.00E-03 2.00E-03 1.00E-03 Fgure3. Velocty on Plate 1 and plate 2 1-4 June 12-14, 2016
14 th Internatonal Users Conference Sesson: ALE-FSI Example 2: Sound transmsson nsde a car Ths smple car example shown n the fgure 4 conssts only of plates welded together and s not a very realstc constructon. A real car would have some sort of a frame, pllars and stffeners. The SEA model has 26 subsystems, 18 steel plates, 6 glass plates and 2 acoustc cavtes. The steel plates and glass plates are modelled as lnear elastc materals. Input power are appled on floor, fender rght front, fender left front and wndsheld. The results are compared wth other code [4] shown n fgure 5 and fgure 6. Fgure4. Smple car SEA model Fgure5. Sound pressure nsde compartment June 12-14, 2016 1-5
Sesson: ALE-FSI 14 th Internatonal Users Conference Fgure6. Velocty of the wndsheld Example 3: Acoustc analyss of a smple shp model Ths smple shp model s courtesy of Numercal Engneerng Solutons, Australa. The SEA model s shown n the fgure 7. Acoustc cavtes Input power at plate 2 Plate 16 Cavty wth water nsde Fgure7. SEA shp on water 1-6 June 12-14, 2016
Spund PressureLevel (db) Velocty (m/s) 14 th Internatonal Users Conference Sesson: ALE-FSI The SEA model has 20 subsystems, 16 steel plates, 3 acoustc cavtes wth ar nsde and one acoustc cavty flled wth water. The steel plates are modelled as lnear elastc materals. Input power s appled on plate 2. The results are compared wth other code [4] shown n fgure 8, 9 and 10.. 5.30E-03 4.30E-03 3.30E-03 2.30E-03 1.30E-03 3.00E-04 Fgure8. Velocty of plate 16 150 145 140 135 130 125 120 Fgure9. Sound pressure level of cavty wth water June 12-14, 2016 1-7
Spund PressureLevel (db) Sesson: ALE-FSI 14 th Internatonal Users Conference 120 115 110 105 100 95 90 Fgure10. Sound pressure level of cavty wth ar (next to the plate 2) Concluson A new SEA solver has been mplemented n. It offers effectve and effcent tools for hgh frequency vbraton and acoustc analyss n complex structures. Three smple examples are ncluded n the paper to show the procedure of runnng SEA analyss wth LS- DYNA. The numercal results are n good agreement wth the analyss usng other code. For the future development, the LS-PrePost wll be updated for user to speed up the generaton of SEA model and to make t more convenent to revew the results. Acknowledgements The authors are grateful to Mr. John Marco of Numercal Engneerng Solutons, Australa for hs nterests and for provdng the shp model (example 3). References 1. Fahy, Frank J. Statstcal energy analyss: a crtcal overvew. Phlosophcal Transacton: Physcal Scences and Engneerng, Vol. 346, No. 431-447, 1994. 2. Rchard H. Lyon and Rchard G. DeJong. Theory and Applcaton of Statstcal Energy Analyss. Cambrdge, MA. 1998. 3. R.S. Langley and K.H Heron. A wave ntensty technque for analyss of hgh frequency vbratons, Journal of Sound Vbraton 159(3), 483-502, 1992. 4. Ennes Sarrad. Statstcal Energy Analyss Freeware. 1-8 June 12-14, 2016