Modeling of Material Damping Properties in ANSYS

Transcription

1 Modelig of Mateial Dampig Popeties i ANSYS C. Cai, H. Zheg, M. S. Kha ad K. C. Hug Defese Systems Divisio, Istitute of High Pefomace Computig 89C Sciece Pak Dive, Sigapoe Sciece Pak I, Sigapoe Abstact A compehesive eview of vibatio dampig i vibatio ad acoustics aalysis is peseted. The teatmet of dampig mateial is a impotat measue fo vibatio ad acoustics cotol i egieeig. The simulatio-based esults o vibatio ad acoustics aalysis ae vey sesitive to the desciptio ad iput methods of dampig popeties. I this pape, the cosideatio of vibatio dampig usig softwae ANSYS fo hamoic ad modal aalysis is addessed. Seveal key poits ae summaized. Itoductio Whe a uacceptable vibatio ad acoustics poblem eeds to be cotolled, it is fistly desiable ad ofte ecessay to udestad its whole atue, such as its oigiatig souce, the atue ad diectio of the vibatio ad acoustics at the poblem locatio, tasmissio path ad fequecy cotet. The, it must be decided whethe the poblem would be best solved by passive o active cotol methods. The passive cotol ivolves modificatio of the stiffess, mass ad dampig of the vibatio system to make the system less esposive to its vibatoy eviomet. This pape is coceed oly with the dampig modificatio i passive cotol methods. If the udesiable vibatio ad acoustics is domiated by oe o moe esoace of the stuctue, it ca be ofte adequately cotolled by iceasig the dampig of the system. Most o-esoat vibatio ad acoustics poblems caot be solved by the dampig teatmet. If a added dampig system is to be effective, the iceased dampig must be sigificatly lage tha the iitial dampig. The most commoly used method of iceasig the dampig is to iclude highly damped polymeic mateial at stategic locatios oto the stuctue. The stuctue ad polyme must iteact with oe aothe i such a way as to cause the polyme to dissipate as much eegy as possible. I pactice, thee ae two kids of dampig teatmets fo vibatio ad acoustics cotol. The fist is called extesioal dampig teatmet. This teatmet is also efeed to as the ucostaied- o fee-laye dampig teatmet. The teatmet is coated o oe o both sides of a stuctue, so that wheeve the stuctue is subjected to flex, the dampig mateial will be subjected to tesio-compessio defomatio. The secod oe is amed as shea type of dampig teatmet. Fo a give weight, the shea type of dampig teatmet is moe efficiet tha the extesioal dampig teatmet. Howeve, this efficiecy is balaced by geate complicatio i aalysis ad applicatio. The teatmet is simila to the ucostaied-laye type, except the viscoelastic mateial is costaied by aothe laye. Theefoe, wheeve the stuctue is subjected to flex, the exta laye will costai the viscoelastic mateial ad foce it to defom i shea. The maximum shea defomatio i the middle laye is a fuctio of the modulus ad the thickess of the costaiig laye, the thickess ad the dampig mateial ad the wavelegth of vibatio i additio to the popeties of the dampig mateial. The actual desciptio of the dampig foce associated with the dissipatio of eegy is difficult. It may be a fuctio of the displacemet, velocity, stess o othe factos. Most of the mechaisms which dissipate eegy with a vibatig system ae o-liea ad cofom eithe to the liea viscous o to the liea hysteetic dampig [1]. Howeve, ideal dampig models ca be coceived which will ofte pemit a satisfactoy appoximatio. I this pape, the categoies of commo dampig mateials i egieeig ae eviewed. Afte that, the model fo desciptio of stuctual dampig is itoduced. Thidly, it is elaboated how ANSYS cosides the dampig popeties fo egieeig pupose. Seveal case studies ae caied out to explai the diffeece of vaious cosideatio methods of mateial dampig effects. Fially, some key poits ae daw fo coect applicatio of dampig effects fo hamoic ad modal aalysis i ANAYS.

2 Categoies of Dampig Mateials They ae seveal types of dampig. Viscous dampig is the fom of dampig that we ae familia with. It is caused by eegy loss that esults fom fluid flow. A example would be the dampig used i vehicle s shock absobes. Fictioal dampig occus whe two objects ub agaist each othe. That is why ou hads get wam whe we ub them togethe. Most of dampig mateials i the maket povided by vaious maufactues belogs to hysteetic dampig. Viscous Dampig [,3] Whe mechaical systems vibate i a fluid medium such as ai, gas, wate ad oil, the esistace offeed by the fluid to the movig body causes eegy to be dissipated. The amout of dissipated eegy depeds o may factos, such as the size ad shape of the vibatig body, the viscosity of the fluid, the fequecy of vibatio, ad the velocity of the vibatig body. I viscous dampig, the dampig foce is popotioal to the velocity of the vibatig body. Viscous dampig foce ca be expessed by the equatio F = cx& (1) whee c is a costat of popotioality ad x& the velocity of the mass show i Figue 1. Figue 1 - A foced damped vibatio of sigle DOF [m=0.5 (kg), k=00 (N/m), c=6(n s/m), F=10 (N)] Whe the sigle spig mass system udegoes fee vibatio, the equatio of motio becomes m & x + cx& + kx = 0 () Assumig a solutio of the fom st x = e, we have the eige o the chaacteistic equatio of the system as ms + cs + k = 0 (3) The solutio of equatio 3 is c c k c k t t t m = m m m m x e Ae + Be (4) whee A ad B ae abitay costats depedig o how the motio is stated. It is obseved that the behavio of the damped system depeds o the umeical value of the adical i the expoetial of equatio 4. As a efeece quality, a citical dampig c c is defied which educes this adical to zeo c k c = 0 o cc = km = mω (5, 6) m m whee ω is the atual cicula fequecy of the system. ω = k m

3 A impotat paamete to descibe the popeties of the dampig is dampig atio ζ, which is a odimesioal atio as c c ζ = = (7) c mω c Based o the value of dampig atio, the motio of the mass i Figue 1 ca be divided ito the followig thee cases: (1) Oscillatoy motio whe ζ < 1. 0 ; () Nooscillatoy motio whe ζ > 1. 0 ad (3) Citical ( ) damped motio whe ζ = I last case, the geeal solutio of the system is x = ω A + Bt e t. Viscous dampig ca be used whateve the fom of the excitatio. The most commo fom of viscous dampig is the Rayleigh-type dampig give by c = α m + β k (8) Coulomb o Fictioal Dampig Coulomb dampig esults fom the slidig of two dy sufaces. The dampig foce is equal to the poduct of the omal foce ad the coefficiet of fictio µ ad is assumed to be idepedet of the velocity, oce the motio is iitiated. Because the sig of the dampig foce is always opposite to that of the velocity, the diffeetial equatio of motio fo each sig is valid oly fo half-cycle itevals. m & x + kx = µ N (9) This is a secod ode ohomogeeous diffeetial equatio. The solutio ca be expessed as x () t k k µ N = Acos t + B si t (10) m m k Hysteetic o Stuctual Dampig I geeal, the dampig mateials ae polymes (sythetic ubbes) which have bee suitably fomulated to yield high dampig capacities i the fequecy ad tempeatue ages of iteest. Whe the mateials ae defomed, eegy is absobed ad dissipated by the mateial itself. The effect is due to fictio betwee the iteal plaes, which slip o slide as the defomatios take place. Whe a stuctue havig mateial dampig is subjected to vibatio, the stess-stai diagam shows a hysteesis loop. Theefoe, the stuctual dampig is also called hysteetic dampig. The aea of this loop deotes the eegy lost pe uit volume of the body pe cycle due to the dampig. To explai the hysteetic dampig, we fist eview the elatioship betwee the espose x ad excitatio iωt foce fo viscous dampig. Fo a hamoic motio, x = Xe, the elatioship betwee them behaves as () t ( ic k)x F = ω + (11) It ca be show i Figue. Equatio 1 gives the eegy dissipated i oe vibatio cycle which is the aea of the loop above π ω 0 ( cxω cosωt + kx siω )( Xω cosωt) dt = πωcx W = Im( F) d Im( x) = (1) whee Im is the imagiay symbol.

4 Figue - The Loop fo Viscous Dampig Fo the hysteetic dampig, similaly, thee is a hysteesis loop to be fomed i the stess-stai o focedisplacemet cuve i oe loadig ad uloadig cycle. It has bee foud expeimetally that the eegy loss pe cycle due to iteal fictio is idepedet of the fequecy fo most stuctual metals, but appoximately popotioal to the squae of the amplitude. I ode to achieve the obseved behavio fom the equatio above, the equivalet dampig coefficiet c eq is assumed to be ivesely popotioal to the fequecy as h c eq = (13) ω whee h is a hysteetic dampig coefficiet. Substitutio of equatio 13 ito 1 esults i the eegy dissipated by the hysteetic dampig i a cycle of motio. W = π hx (14) Model of Dampig Popeties[5] Stuctual dampig facto γ Beside the viscous dampig coefficiet c, hysteetic dampig coefficiet h ad the dampig atio ζ, thee is aothe vey impotat paamete, stuctual dampig facto, to descibe the popety of the dampig mateiel. The foced motio equatio of a sigle spig mass system with a hysteetic dampe is () t m& x + c x& + kx = f (15) eq Fo a hamoic poblem, it becomes ω mx + k 1 i ω ω ζ eq x = f () t (16)

5 whee ζ eq ceq h = =. c mω ω c Fo the modal dampig, ω = ω, theefoe, we have + k( 1 iγ ) x f (t ) m & x = (17) whee γ = ζ = h k is called the stuctual dampig facto o modal dampig atio. eq Fo the viscous dampig, similaly, the viscous dampig facto is γ = ζ. Complex Stiffess The effect of polyme mateial o the dampig of the whole stuctue is iflueced by the mateial stiffess as well as by its dampig. These two popeties ae coveietly quatified by the complex Youg s modulus E ( 1 iη ) E o the complex shea modulus G( 1 i η ) G. η G ad η E ae usually assumed to be equal fo a give mateial. Whe the mateial is subjected to cyclic stess ad stai with amplitude σ 0 ad ε 0, the maximum eegy stoed ad dissipated pe cycle i a uit volume ae as Maximum eegy stoed pe cycle = E (18) ε 0 Eegy dissipated pe cycle = π Eη ε 0 (19) Compaed to equatio 17, the complex stiffess k( i h k 1. Defiig the loss facto = h k G( iη) 1 ) is simila to the complex modulus E( iη) η, the complex stiffess ca be expessed as k( iη) 1 o 1. η may 5 vay fom fo 10 pue alumium to 1.0 fo had ubbe. The stuctual dampig facto γ is equivalet to loss facto η. Loss facto is a tem used to quatify dampig pefomace. A physical itepetatio of the loss facto ca be obtaied as follows. The eegy dissipated pe cycle fo a stuctual damped system is 1 W = π hx = π ηkx = π η kx = π ηu m (0) whee U m is the maximum stai eegy stoed. Theefoe, we have 1 W 1 eegy disspated pe cycle η = = π π maximum stai eegy U m Fom equatio 1, it is foud that the loss facto is a way to compae the dampig of oe mateial to aothe. It is a atio of the amout of eegy dissipated by the system at a cetai fequecy to the amout of the eegy that emais i this system at the same fequecy. The moe dampig a mateial has, the highe the loss facto will be. The method of epesetig the stuctual dampig should oly be used fo fequecy domai aalysis whee the excitatio is hamoic [4]. (1) Liea Mathematical Model [5] May oliea aalyses of damped espose of stuctues have bee caied out usig aalytical epesetatios of such a hysteesis loop as ν 1 σ = E ε [( ε + ε ) ε ] 0 0 ()

6 s ν 1 σ = E ε + [( ε ε ) ε ] 0 0 (3) whee σ is the stess duig the loadig pat of the cycle ad σ s is that duig the uloadig pat. ν is Poisso s atio. ε 0 is the iitial stai. is the odimesioal paamete i desciptio of hysteesis loop. A alteate fom, somewhat simple, is v w σ = ( ε ) ε ± η( ε ) 1 ε ε 0 ε E 0 β with ( ε ) = E ( 1+α ε ) E (4) Oe of the best kow epesetatios of the state equatio is kow as the stadad liea model, ad it gives the followig elatioship betwee stess ad stai: dσ dε σ + a = Eε + be (5) dt dt This paticula equatio epesets a moe complex elatioship betwee stess ad stai tha eithe dε Hooke s law σ = Eε o the simple dashpot spig combiatios, fo which σ = Eε + be. Two dt paametes a ad b i equatio 5 ae the costats of stess ad stai elaxatio espectively. If the i t i t applied stess ad stai vay hamoically, of the type σ = σ e ω ad e ω 0 ε = ε o, the the equatio gives 1 iωb σ = Eε 0 1 iωa 1+ ω ab b a σ = E iω ε (6, 7) 1+ ω a 1+ ω a ; Fom equatios 6 ad 7, we ca see that loss facto is usually depedet o the fequecy. The modulus is also fequecy depedece if the costat of stess a is ot zeo. Vibatio Dampig i softwae ANSYS [6] The dampig matix C i ANSYS may be used i hamoic, damped modal ad tasiet aalysis as well as substuctue geeatio. I its most geeal fom, it is: N mat N ele [ C] = [ M ] + β[ K ] + β [ K ] + β [ K ] + [ C ] + [ j j c ζ α C ] (8) j= 1 k = 1 k whee α β β j β c ζ costat mass matix multiplie (iput o ALPHAD commad) costat stiffess matix multiplie (iput o BETAD commad) costat stiffess matix multiplie (iput o MP, DAMP commad)-- mateial-depedet dampig. It is oted that diffeet dampig paametes ae defied fo diffeet types of aalysis whe usig the mateial-depedet dampig. Fo example, MP, DAMP i a spectum aalysis specifies a mateial-depedet dampig atio ζ, ot β. vaiable stiffess matix multiplie: (available fo the hamoic espose aalysis, is used to give a costat dampig atio, egadless of fequecy) ζ ζ η β = = = (9) c πf ω ω costat dampig atio (iput o DMPRAT commad). Fom equatio 9, the dampig atio ζ should be η whee η is the loss facto.

7 f fequecy i the age betwee f b (begiig fequecy) ad f e (ed fequecy); [C ζ ] fequecy-depedet dampig matix [C ζ ] may be calculated fom the specified ζ (dampig atio fo mode shape ) ad is eve explicitly computed. { u } f ζ T { u } [ C ]{ u } 4πf ζ ζ = (30) the th mode shape fequecy associated with mode shape ζ = ζ + (31) ζ m costat dampig atio (iput o DMPRAT commad) ζ m modal dampig atio fo mode shape (iput o MDAMP commad) [C k ] elemet dampig matix Rayleigh Dampig α ad β The most commo fom of dampig is the so-called Rayleigh type dampig [ C] α [ M ] + β[ K] advatage of this epesetatio is that the matix becomes i modal coodiates =. The C = α I + βλ (3) C is diagoal. So fo the th mode, the equatio of motio (equatio 15) ca be ucoupled. Each oe is of the fom ( α + βω ) q& + ω q Q q & + = (33) Let ζ ω = ( α + βω ) (34) m The equatio 33 educes to whee q & + ζ ω q& + ω q = Q (35) m ζ m is the th modal dampig atio. The values of α ad β ae ot geeally kow diectly, but ae calculated fom modal dampig atios, ζ m. It is the atio of actual dampig to citical dampig fo a paticula mode of vibatio,. Fom equatio 34, we have ζ m α β = + ω (36) ω I may pactical stuctual poblems, the α dampig (o mass dampig) which epesets fictio dampig may be igoed (α = 0). I such case, the β dampig ca be evaluated fom kow values of ζ m ad ω which epesets mateial stuctual dampig. It is oted that oly oe value of β ca be iput i a load step, so we should choose the most domiat fequecy active i that load step to calculate β. I case whee the dampig popeties vay cosideably i diffeet pats of the stuctue, the above techiques caot be used diectly. A example is the aalysis of soil-stuctue iteactio poblems, whee thee is sigificatly moe dampig i the soil tha i the stuctue.

8 Mateial-depedet Dampig β j It specifies beta dampig β j as a mateial popety (iput o MP, DAMP commad). It is oticed that MP, DAMP i a spectum aalysis [ANTYPE,SPECTR] specifies a mateial-depedet dampig atio ζ i, ot β j. Costat Dampig Ratio ζ The commad DMPRAT is used to epeset the loss facto whe ζ is set to be η. It is specified as a decimal umbe with the DMPRAT commad ad is the simplest way of specifyig dampig i the stuctue. It epesets the atio of actual dampig to citical dampig. DMPRAT is available oly fo spectum, hamoic espose, ad mode supepositio tasiet dyamic aalyses. As stated i equatio 9, the costat dampig atio is used to calculate β c. Modal dampig ζ m It is specified with the MDAMP commad ad gives us the ability to specify diffeet dampig atios fo diffeet modes of vibatio. MDAMP is available oly fo the spectum ad mode supepositio method of solutio (tasiet dyamic ad hamoic espose aalyses). Togethe with equatios 30 ad 31, ζ m is used to compute the fequecy depedet dampig matix [C ζ ]. Elemet dampig [C k ] Elemet dampig ivolves usig some special elemet types havig viscous dampig chaacteistics, such as COMBIN7, COMBIN14, COMBIN37, COMBIN40 ad so o. Elemet dampig is applied via elemet eal costat. Vibatio Dampig fo Hamoic Aalysis Seveal discete vibatio systems have bee used fo checkig how to coside the dampig popeties whe usig ANSYS fo hamoic aalysis. Viscous Dampig A sigle mass spig system (efeig to Figue 1) By the method of eal costat of elemet (Figue 3)

9 Figue 3 - Respose of Viscous-damped Sigle DOF System Diectly iput the viscous dampig coefficiet c as eal costat of the elemet (COMBIN14). By the method of Rayleigh Beta dampig (iput o BETAD commad) β = ζ ω (Figue 4) Calculate the dampig atio ζ which depeds o the atual fequecy of the system. The compute the paamete β with fomula β = ζ ω. Figue 4 - Respose of Viscous-damped Sigle DOF System (β=0.03s)

10 By the method of mateial popeties (iput o MP, DAMP commad), β = ζ ω (Figue 5) j j Simila to the iput of the paamete β above, the dampig popeties of the damped stuctue ca be defied via assigig the elated elemet (COMBIN14, fo example) o a specific dampig value as the mateial popeties [MP,DAMP]. Figue 5 - Respose of viscous-damped sigle DOF system (β j =0.03s) Multiple mass spig system (efeig to Figue 6) Figue 6 - A foced two DOFs of damped vibatio (figue6.jpg) [m 1 =m =0.5 (kg), k 1 =k =k 3 =150 (N/m) c 1 =5(N s/m), c =0.1(N s/m), c 3 =0.05(N s/m) F 1 =10 (N), F =500 (N) ] By the method of eal costat of elemet (Figue 7) It is staightfowad to specify the viscous dampig coefficiets ci of each dampe via eal costats of the elemets used, espectively.

11 Figue 7 - Respose of Viscous-damped Two DOF System By mateial popeties (iput o MP, DAMP commad ), β = ζ ω. j j It is eeded to calculate the atual fequecies of the system ad defie the most sigificat mode fo the poblem at had because thee ae moe tha oe atual fequecy fo the multiple-dof vibatio system. Say the domiat o majo atual fequecy be ωd, the mateial-depedet dampig βi ca be calculated espectively with this atual fequecy. It is expected that it is had to specify the domiat atual fequecy with icease of the umbe of degee-of-feedom of vibatio system. Figue 8 shows the esposes of two masses whe the fist atual fequecy ω 1 =17.3(Hz) is used as domiat fequecy. Figue 9 shows the espose of two masses whe the secod atual fequecy ω =30.00(Hz) is used as domiat fequecy. It is see that obvious discepacy is obseved whe a appopiate atual fequecy is selected as the domiat fequecy as show i Figue 9. It is oted that oly oe value of β ca be iput i a load step, theefoe, the method by Rayleigh Beta dampig ca ot be used diectly to specify the dampig popeties of the vibatio system with diffeet dampig mateials. Figue 10 shows the espose of two masses whe β=

12 Figue 8 - Respose of viscous-damped two DOF system (β 1 =0.033, β = , β 3 = ) Figue 9 - Respose of viscous-damped two DOF system (β 1 =0.011, β =. 10-4, β 3 = )

13 Figue 10 - Respose of Viscous-damped Two DOF System (via Rayleigh dampig method: β= ) Hysteetic Dampig β c via costat dampig atio ζ (iput o DMPRAT commad) As metioed above, the commad DMPRAT ca be used fo the hysteetic dampig iput fo sigle degee of feedom of vibatio system. Agai fom equatio 9, the dampig atio ζ should be η if the loss facto is η. Figue 11 shows the espose of sigle DOF of vibatio system with a hysteetic dampe. β j via mateial depedet dampig (iput o MP, DAMP commad) Mateial-depedet dampig β ca be used fo specifyig the hysteetic dampig popeties both j i sigle ad multiple DOF of vibatio systems. The β should be calculated with equatio 9 if j the loss factos η j is kow. ( ω ) η j β = (37) j ω whee η j (ω) is the fequecy-depedet loss facto of the j th mateial. ω is the cicula fequecy.

14 Figue 11 - Respose of Hysteetic-damped Sigle DOF System [m=0.5 (kg), k=00 (N/m), η=0.6] Fo hamoic vibatio aalysis, equatio 37 should be applied fequecy by fequecy. I othe wods, the dampig β eeds to be give fo each aalysis cicula fequecy, espectively. j Figue 1 shows the esposes of two DOF of vibatio systems (Figue 6) with thee hysteetic dampig dampes. Figue 1 - Respose of Hysteetic-damped Two DOF System (Via mateial-depedet popeties method: η 1 =0.5, η =0.3, η 3 =0.)

15 Vibatio Dampig fo Modal Aalysis Modal aalysis is used to detemie the vibatio chaacteistics (atual fequecies ad mode shapes) of a vibatio system. The esults fom modal aalysis may be futhe applied fo aothe dyamic aalysis via mode supepositio method. Alpha (mass) dampig, Beta (stiffess) dampig, Mateial-depedet dampig atio (iput o MP, DAMP commad) ad elemet dampig (applied via elemet eal costat) ca be specified i modal aalysis. It is oticed that i the mode supepositio method, oly Rayleigh o costat dampig is allowed, ad explicit elemet dampig i such elemets as COMBIN 14 is ot allowed. Cosideig equatios 30 ad 31, the modal dampig, ζ, is the combiatio of seveal ANSYS dampig iputs as α β ζ = + ω + ζ + ζ ω m whee α is uifom mass dampig multiplie (iput o ALPHAD commad); β uifom stiffess dampig multiplie (iput o BETAD commad); ζ is costat dampig atio (iput o DMPRAT commad) ad ζ is the modal dampig atio (iput o MDAMP commad). m Fo the vibatio stuctue cosistig of sigle mateial oly, mateial-depedet dampig, β j, povides the same esults as specified Beta (stiffess) dampig, β, sice β = ζ ω. It is see that the fequecydepedet dampig teatmet such as hysteetic dampig i equatio 37 is ot applicable fo modal aalysis. I pactice, Rayleigh dampig paametes of cotiuum dampig mateials ca ot be ead fom the vedo s dampig data sheet (modulus ad loss factos). Theefoe, Rayleigh dampig may be oly suitable fo the vibatio system with sigle degee of feedom. Summaies Thee ae seveal methods to iclude the mateial dampig i ANSYS. The followig poits ae daw out fo usig dampig i ANSYS: Elemet elated dampig iput is oly suitable fo some special elemets such as COMBIN7, COMBIN 14, COMBIN37, COMBIN40 ad so o. The viscous dampig coefficiets ca iput by the eal costats diectly. It is suitable fo sigle ad multiple DOF vibatio systems; Rayleigh dampig (iput o BETAD ad ALFAD commads) is suitable fo sigle DOF vibatio system because it depeds o the domiat atual fequecy ad dampig atio. Fo multiple DOF systems ad cotiuum vibatio system, it is difficult to idetify the domiat atual fequecy ad modal dampig atio. β = ζ ω if α is assumed to be zeo; Costat dampig atio (iput o DMPRAT commad) is used to specify the hysteetic dampig diectly fo a sigle DOF of vibatio system ad/o the system cosistig of oly oe kid of mateial. The iput costat dampig atio is half of the loss facto. ζ = 0. 5η ; Mateial depedat dampig (iput o MP, DAMP commad) ca be used to iput the viscous dampig β = ζ ω ad/o hysteetic dampig β = η ( ω ) ω. With the simila limitatio of j j j i Rayleigh dampig, it ca oly used to epeset the viscous dampig of sigle DOF vibatio system ad/o the cotiuum system cosistig of oly oe mateial. Howeve, it ca be used to epeset hysteetic dampig of multiple DOF system ad othe multiple-mateial cotiuum vibatio system ude hamoic aalysis based o each aalysis fequecy. Obviously, this method is ot applicable to damped modal aalysis. Fo damped modal aalysis, all fou methods above to epeset the mateial dampig ca be used with coespodig applicatios. Rayleigh Beta dampig is oly applicable fo the atual fequecy aalysis of sigle DOF vibatio systems i pactical use. Costat dampig atio (iput o DMPRAT commad) is applicable fo modal aalysis of the cotiuum vibatio system with j j j (38)

16 oly oe kid of hysteetic dampig mateial. Elemet elated dampig iput methods via eal costats ca be used fo modal aalysis of both sigle ad multiple DOF discete vibatio systems with viscous dampig popeties. The special attetio should be paid whe we use ANSYS fo damped vibatio ad acoustic aalysis. Refeeces 1) Deys J. Mead, Passive Vibatio Cotol, Joh Wiley & Sos, Ic., 000 ) Sigiesu S. Rao, Mechaical Vibatios (3 d editio), Addiso-Wesley Publishig Compay, ) William T. Thomso, Vibatio Theoy ad Applicatios, Petice-Hall, Ic, Eglewood Cliffs, N.J, ) Mauice Petyt, Itoductio to Fiite Elemet Vibatio Aalysis, Cambidge Uivesity Pess, ) Ahid D. Nashif, Davis I. G. Joes ad Joh P. Hedeso, Vibatio Dampig, A Wiley- Itesciece Publicatio, Joh Wiley & Sos, Ic., ) O-lie ANSYS Theoy Refeece, Release ) Roy S., Reddy J, N., Fiite-elemet models of Viscoelasticity ad diffusio i Ashesively Boded Joits, Iteatioal Joual fo Numeical Methods i Egieeig, 6(1988) ) Yi S., Ahmad M. F., Hilto H. H., Fiite elemet algoithms fo dyamic simulatios of viscoelastic composite shell stuctues usig cojugated gadiet method o coase gaied ad massively paallel machies, Iteatioal Joual fo Numeical Methods i Egieeig, 40(1997): ) Yi S., Lig S. F., Yig M., Hilto H. H., Viso J. R., Fiite elemet fomulatio fo aisotopic coupled piezoelecto-hygo-themo-viscoelasto-dyamic poblems, Iteatioal Joual fo Numeical Methods i Egieeig, 45 (1999) ) Chug P. W., Tamma K. K., Nambuu R. R., A fiite elemet themo-viscoelastic ceep appoach fo heteogeeous stuctues with dissipative coectos, Fiite Elemets i Aalysis ad Desig, 36 (3-4) , 000