Evaluation of damping in dynamic analysis of structures

Transcription

1 Evaluaton of dampng n dynamc analyss of structures Tepes Onea Florn, Gelmambet Suna Abstract From physcal pont of vew, the dumpng represents the sol sesmc exctaton energy taen over process through nternal absorpton, rubbed between exstent layers, as cracs on rocy foundatons Generally, on heavy dams dynamc analyss t s consdered a vscous dump, proportonal wth deformaton speed. The dumpng can be evaluated on expermental bases or on envronmental condtons measurements. The latest determne hgher values of dumpng elements. Ths t could be explaned wth the local factors nfluence whch s not possble to modeled as baclash treatment, foundaton ground characterstcs, the concrete technology. Ths represents an atypcal dsspate phenomenon. A major nfluence s done by the exctaton level as real sesm or expermental exctaton. The present wor s about to establsh the nfluence of the dsspate effect of the baclash on concrete blocs. The baclash fnte elements modelng mae ths possble, studyng dfferent stuatons as rub effect, coheson effect, sesmc acton on varyng drectons wth the same accelerogram of.4g. The studed blocs have the same dmensons, the relatve dsplacement beng obtaned by foundaton stffness modfed under two bloc parts. Keywords dsspaton, dumpng exctaton, raylegh, spatal mesh. I. RAYLEIGH MODEL Internal energy dsspaton nsde the structure s caused by nternal phenomena n rollng and hysteretc dampng. Phenomenon s essentally nonlnear, a lnearzaton s necessary to ntroduce these effects. Expermentally observed that the energy dsspated by hysteretc dampng cycle s ndependent of frequency for dfferent materals. On the other hand the vscous dampng the energy s proportonal to the frequency. On the other hand the vscous dampng the energy s proportonal to the frequency. In ths case you can use a smple model, the dampng coeffcent of the materal s gven by a coeffcent of vscosty on the frequency. Solvng by usng the fnte element method, dampng matrx Manuscrpt receved May 9, : Revsed verson receved May 9. łepes Onea Florn s wth the Faculty of Cvl Engneerng,Unversty Ovdus of Constantza (correspondng author to provde phone: ; fax: ; e-mal: tflorn@unv-ovdus.ro). Gelmambet Suna s wth Faculty of Cvl Engneerng,Unversty Ovdus of Constantza (correspondng author to provde phone: ; fax: ; e-mal: gelmambets@yahoo.com). [C] structure can be syntheszed as mass matrx dampng elements n arrays: ( ) T [ C ] = c[ N] [ N] dv v where C s the dstrbuted vscous dampng. Yet determnng scalar c s untenable. Thus deprecaton s typcally based on the fracton of crtcal dampng, determned expermentally or smlar structures. Therefore matrx [C] of structure s not generally assembled from arrays of dampng elements, but s bult usng mass and stffness matrces of the entre body of evdence, together wth results on sze exprmentale dampng. Raylegh showed that the dampng matrx form C = α M + βk, where α and β are scalng constants, satsfes the orthogonalty condtons. general expresson: () You can use more N l C ] = [ M ] al ([ M ] [ K]) () l= [ where N equals the number of degrees of freedom than the structure. Is easly seen that expresson s obtaned from the expresson for N =. Scaled multplers determne the fracton from the crtcal dampng γ. Raylegh model, complete or smplfed, has the advantage that t does not ntroduce couplng between modes of vbraton of the structure. Ther shapes are orthogonal mass matrx and stffness matrx and therefore the dampng matrx expressed by ths model. Thus dampng matrx allows decouplng of moton equatons. ν a / a ν = + a / a.. 4. (Hz) Fg. Relatonshp between parameters α and β scaled model of dampng of Raylegh and fracton of crtcal dampng ν Deprecaton s the sum total of the amortzaton structure of Issue, Volume 4, 4

2 each vbraton mode. Deprecaton of each mode of vbraton can be observed, for example by mposng proper ntal condtons that specfcally measures the vbraton ampltude free vbraton wth dampng. An mportant factor s that we have the ablty to measure the dampng rate ξ. Dampng rato ξ n step-by-step ntegraton should always be nown. In ths case t s necessary to evaluate the dampng matrx C explctly, the matrx used to determne the dampng raton ξ. Example: Consder a system wth multple degrees of freedom: = and =, these two modes havng two crtcal dampng: % and % and ther correspondng dampng factors ξ =. and ξ =.. The objectve s to determne the constants a and b for Raylegh type dampng, for the ntegraton step by step for all values. T φ ( αm + βk) φ = ξ = j α + β ξ α + 4 β =.8 So you get to see dfferent pars of coeffcents α and β pars accordng and. Procedure for calculatng the parameters α and β n the above example may suggest usng a more complcated dampng matrces f we have more than two dampng ratons that are used to determne the matrx C. An mportant observaton s that f N>, the dampng matrx s n general a full matrx. Cost analyss wth a dampng matrx s not band tme when ntegraton s hgh. One dsadvantage s that Raylegh dampng hgher modes of vbraton modes are amortzed over low, for each Raylegh constant that s selected. Raylegh coeffcents n practce, for a specfc structural analyss, are used usng nformaton taen may be selected from a smlar structure. Coeffcents α and β values depend on the energy dsspated by the structure feature. In dscusson we consder the dampng characterstcs of structures that can be represented both n proporton. Deprecaton usng superposton method as well as n drect ntegraton. In many analyss s consdered to exst Deprecaton proportonal, but wth varyng materal propertes for structural analyss s used dsproportonate deprecaton. II. EVALUATION BY AMPLIFYING THE RESONANCE DAMPING Ths procedure used to evaluate dampng s based on the observaton that a harmonc response as a result of applcaton of harmonc exctatons on the structure, the frequences and ampltudes prescrptons. Wth such equpment the frequency response curve for the structure can be bult usng a harmonc load p o sn t a small band of frequency around the resonance frequency, thus resultng dsplacement ampltude frequency appled. Dynamc amplfcaton factor s the ampltude response for a partcular frequency shfts reported n response to appled statc, beng nversely proportonal to the dampng rate: D β = = () ξ When the statc response and the response to resonance are denoted by p and p β = when dampng rate s gven by: ξ = pmax (4) where β s the frequency. In practce, always, t s dffcult to apply exactly the resonance frequency, but t s convenent to determne the maxmum ampltude response p max s obtaned for one am that low frequency. In ths case t s obvous that the dampng rate can be evaluated as follows: p o p max p o p po ξ = = (5) p p p o D max p p max = ξ p max β β ξ = β β β Fg. Evaluaton by amplfyng the resonance dampng Error that appears n equaton 5 results n neglectng the dfference between amortzed and outstandng frequences, but s nsgnfcant for common structures Ths method of analyss of dampng requre smple nstrumentaton capable of measurng the relatve ampltudes of the dsplacements. Always statc evaluaton dsplacement can be problematc for there are many types of loadng systems. III. BANDWIDTH METHOD Issue, Volume 4, 5

3 Is evdent from the general expresson of response p dsplacements, p = ( ) ( ) β + ξβ where β s represents the frequency response shape s controlled dampng system, dampng rate s then derved. The band method, deprecaton rate s determned by the p = t s common for the nput ampltude s half the resonance amplfcaton. frequency response s reduced at ( ) β p = p ξ ( β ) + ( ξβ ) Or rasng to square both sdes: = 8ξ ( β ) + ( ξβ ) (7) and frequency rate s gven by: neglectng the ampltude are: β β (6) β = ξ ± ξ + ξ (8) ξ two frequences correspondng to half = ξ ξ = + ξ ξ β = ξ ξ (9) β = + ξ ξ () statc response. Always need to be traced accurately the frequency response curve. IV. ENERGY LOST IN A CYCLE (TEST THE RESONANCE) If applances that are avalable to measure the phase dfference between appled force and resultng dsplacements, dampng can be evaluated only by a smple test just to resonate, not necessary to buld the frequency response curve. Procedure nvolves determnng resonance frequency by adjustng the nput untl the response s a 9 phase dfference from the force appled. Therefore appled load s exactly balanced by the dampng force. If the structure has a lnear vscous dampng, the curve wll be an ellpse (fg.4). In ths case, the dampng coeffcent may be determned drectly from the report maxmum dampng force at maxmum speed: f D,max p c = = () v ɺ max where the maxmum speed s gven by the product of frequency and ampltude of movement. If vscous dampng s not lnear dsplacement dagram wll be ellptcal. p The dampng rate s gven by half the dfference between the two frequences: ξ = ( β β ) () Ths method for assessng the dampng raton s shown n fgure. Horzontal lne was drawn at a value equal to the pea at resonance ( ). 4 f p max f =.87 f f ξ = =.8% f f + f f f + f max = = Fg. Bandwth method f (Hz) The dfference between the two frequences obtaned by a horzontal lne wth ntersecton response curve s twce the dampng raton. It s obvous that ths s the technque to avod Fg.4 Energy dsspated n a cycle Vscous dampng coeffcent can be defned as havng lost energy same cycle as seen n the force-dsplacement dagram. Amortzaton assocated wth equvalent vscous forcedsplacement dagram s the same area and same maxmum dsplacement of the force-dsplacement dagram. In ths case the dotted lne fg.4 s equvalent to the contnuous lne. In ths case the ampltude of the appled force s gven by: D p = () π p Where D s the area wthn the force-dsplacement dagram, representng the energy lost per cycle. Substtutng ths n expresson s obtaned for the equvalent vscous dampng coeffcent by the energy lost per cycle: D ceq = (4) π p Issue, Volume 4, 6

4 In many cases t s easer to defne deprecaton through crtcal dampng coeffcent. Defnng a measure of crtcal dampng coeffcent s the mass and frequency terms: c c = (5) Force-dsplacement dagram obtaned n ths way wll be as shown n fg.5 f the structure s lnear elastc. whch s defned by the dampng force. Ths forcedsplacement relatonshp can be expressed as follows: vɺ f D = ζ v (9) vɺ where ξ s the dampng coeffcent hysteretca. Dagram for a force-dsplacement cycle s presented fg.6. Fg.5 Force dsplacement dagram Rgdty s shown by curve angle. Alternatvely, rgdty may be expressed by area under the force-dsplacement dagram as follows: wd = (6) p So dampng rate can be acheved by combnng equatons 4,6. c wd ξ = = (7) c 4π w Dampng rate defned by equaton 7 s apparently ndependent of frequency, t depends drectly on the energy lost per cycle correspondng to maxmum dsplacement. Always, for any mechansm of vscous dampng energy lost n the system wll be proportonal to the frequency. Alternatvely when the dampng rate s evaluated by test of reasonng, vscous dampng coeffcent s obtaned by substtutng eq. 4 to 7. Thus resultng dampng coeffcent nversely wth frequency: 4wD ceq = ξ (8) p whch demonstrates agan that vscous dampng s dependent on frequency. V. HYSTERETIC DAMPING Although the dampng mechansm results n a convenent form for equaton moton, expermental results seldom match ths pattern. In many practcal cases vscous dampng concept defned by the energy lost per cycle produces a reasonable approxmaton to the results of experments. A mathematcal model wth the property that s ndependent of frequency dampng s provded by the concept of deprecaton hysteretca c s Fg.6 Hysteretc dampng force It s noted that dampng resstance has the same effect wth the ncreasng dsplacement lnear elastc forces, but the meanng s reversed dampng forces when the dsplacements decrease. Hysteretca energy lost n a cycle on ths mechansm s: w D = ζ p () If ths energy s lost hysteretca represented by equvalent vscous dampng, vscous dampng rate s gven by equaton 7. In other words equaton (7) can be used to express the structure dampng rate regardless of the mechansm of energy loss. By substtuton ec. and ec.6 n ec.7, hysteretca dampng coeffcent can be expressed as follows: ζ = π ξ () It s clear that that hysteretca dampng s ndependent of frequency at whch the test was made to contrast wth vscous dampng coeffcent presented n ec.8. VI. INTRODUCTION The dampng matrx s obtaned from the Cauchy sequence: p C = M a M K () = Where the coeffcents smultaneous equatons: ( ) a ξ = + a + a a a =,..p are obtan from p p p For p=: C = αm + β K () Issue, Volume 4, 7

5 Where α and β are constants that can be obtaned from two dumpng ratos of two dfferent frequences. Ths study s made for an dealzed symmetrc concrete dam It was used two calculus models for the dam-foundaton ensemble; a plan one and spatal model wth smultaneous calculaton. VII. PROBLEM FORMULATION The plane fnte element mesh s made by 8 quadrlateral elements for the foundaton and 56 elements for the dam. The elastcty modulus for the dam was pc E = dan / cm and for the foundaton b E f = 5 /, wth the dam s hgh of m and λ dan cm λ the slope = =,5.The dam s made by two plots 5m wdth each other separate for a baclash of mm. The two plots adjacent nodes, correspondng wth the space, have the same quota on x and z axs. Ths nodes can be connected wth the help of sprngs n order to model the frcton.we can notce that n the case of plane mesh the fundamental vbraton mode s flexural on upstream-downstream, the second mode s flexural too but on the hgh of the dam, the others modes are of torson. VIII. PROBLEM SOLUTION Between the two consdered bloc parts s a relatve moton made evdent by superor mode shapes. The phenomenon s more complex because on the separaton dam parts surface t appear also the rub and stre. Ths study ponts out only the rub phenomenon. For the smulaton of the rub energy dsspaton, the nterface nods were connected wth Truss fnte elements. These elements wthstand to the bloc parts relatve moton worng le sprngs. The relatve moton phenomenon s due to the dfference n phase result of dfferent hgh, modfed exctaton and dfferent structural propertes. For relatve moton between dam parts calculus smplfy, the foundaton elastcty modulus was changed. In comparason, a numercal ntegraton wthout sprngs lmtaton was done. The results are presented n the followng fgures. The rub between the dam parts s unform dstrbuted on baclash surfaces. Ths dstrbuted force s consdered hypothetcally concentrate n nods. As t was presented, Truss elements model the rub phenomenon. Changng the sprngs stffness n accord wth feedbac structural response dfferent results were obtaned as are presented n followng table (for a crownng node). It s notced that the structural response s almost dentcal for a large nterval of the sprngs stffness. It was chosen a.m sprngs area. It s notced that rub force could not overtae a lmt value, and the x drecton maxmal dsplacement of the crownng node number become.49e- n comparason wth.48e- whch s the value correspondng to no baclash energy dsspaton hypothess. These dsplacements are measured compared to reference base. For the spatal mesh the frst vbraton mode mples a symmetrcal dsplacement and flexural on upstreamdownstream drecton of the two plots, and for the second mode a antsymmetrcal dsplacement. To start wth the 6 s vbraton mode t appear also a rotaton of the two plots, mples a relatve movng of the plots surfaces n the spae between them. The propose of ths wor s to study the effect of the superor vbraton modes on the energy dsspaton n the baclash between the two plots.because both masses and stffness matrx are orthogonal, dampng matrx s orthogonal too. From orthogonal condton we obtan: T φ ( αm + β K) φ = () Where, j j φ φ are egen vectors, s crcular frequency, ξ fracton of crtcal dumpng. The equaton become: α + β = ξ. For determne the α and β coeffcents nfluence, t was made a parametrc study for plane and spatal dam- foundaton dscrete mesh. Crtcal damp fracton was too as constant ξ =,5 for whole vbraton modes because of the fact that massve structure as a concrete dam s, t s possble to obtan, after the structure exctaton (wth a value lower that the sesmc value), only the fracton of crtcal dumpng correspondng to the frst vbraton mode. The calculus was made n both cases of fnte elements, for the frst vbraton modes. If we couple + and solvng the equaton systems obtaned result α and β coeffcents. So for plane dscrete mash α =, 44, β =,5E for and and α =.97, β = 5.66E 4 for and. We can observe that the effect of the mass matrx ncrease and the effect of stffness matrx decrease n the same drecton wth the ncrease of the second frequency tae nto account.in spatal mesh case, the fundamental vbraton mode s reduce rad s =, 48 / and for the plane mesh =. rad / s. Ths dfference appears because n spatal mesh we tae nto account the torson vbraton modes also.for the spatal mesh and for the frequency and α=,89 and β =, 5E whle f use the frequency and 9 α =, 6 and β =.56E. In the case of spatal dscrete mesh we can notce a mass matrx nfluence grow and a stffness matrx nfluence dmnuton n the same tme as the pulsaton value grow. The varaton of the α and β factors s much reduce when s use the spatal mash. For the Issue, Volume 4, 8

6 spatal mesh case we obtan a mean value α Ed =,99 and for α =. All the results are presented n the plane mesh med.7 tables and Table Plane mesh Num. Ω (rad/s) α β +,44,59E- +5,7 9,7E-4 +7,78 7,67E-4 +,97 5,66E-4 Table Spatal mesh Num. 7,44 6,49 4 4,8 5 7, 6 4,7 7 45, ,8 9 5,5 + α β +,89,5E- +5,989,97E- +7,4,689E- +9,6,567E- The calculus was resume for a crtcal dump fracton 8% n wtch case we obtan for spatal mesh and the pears and 9 the followng results α=.69 and β=.5e-, results wth no bg dfference compare wth the case of crtcal dump fracton of 5%. It s obvous that only for a spatal dscrete mash the obtaned results are close to realty. The nfluence of superor modes use n the case of spatal mesh have no sgnfcant effect on the α and β coeffcents as t presented n table. It s notced that n the same tme wth the ncrease of the frequency the mass matrx effect ncrease to and also the stffness matrx effect decrease. So, we can say that the stffness matrx effect connected wth frequency s major. After coeffcent calculus, t was analyzed the dam response at the same exctaton wth and wthout dampng matrx effect. It was followed the effect of usng vbraton modes -, -5,-7,- n mass and stffness matrx coeffcents on the stress and dsplacement response. In the table,4,5,6 are presented stress and dsplacement values for dfferent coeffcent pars α and β, for plane and spatal mesh. Table. Plane mesh-dsplacements compare (node 6) Wthout.78E-m dapng +.4E E E E- Table 4 Plane mesh-stress compare σ(-)dan/cm Wthout -8. dapng Table 5 Spatal mesh-dsplacements compare Wthout.4E- dapng +.9E- +.6E E E- Table 6 Spatal mesh stress compare Wthout -. dapng Issue, Volume 4, 9

7 + -9. Table 7 dsplacement comparatve values for the spatal mesh for two crtcal dumpng ratos of 5% and 8%. (node ) 5% Raylegh 8% Rayleg Dference % (x)-.878e- (y)-.8799e- (z) -.4E- -.6E E- -.7E The dsplacement comparatve graphcs are presented n fgures and and the stress calculus ponts n fgure. Fgure 9 β ( ) for plane mesh and spatal mesh It s also notce that as well as for plane and spatal dscrete mesh, f dampng matrx s used, the stress and efforts values are almost smlar for all the coeffcent pars α and β used. It was also notced that for Raylegh models use, only the frst vbraton modes are requred. Major response dfferences of 4% are obtaned only between crtcal dams of 5% and 8%. Fgure Modal analyze. Spatal mesh for a concrete dam. The sx vbraton mode Freq=4.7rad/s; T=.49s Fgure 7 Dsplacement comparaton for spatal mesh Fg. Modal analyss of gravty dam mesh flat; Freq.. rad/s.the frst vbraton mode. IX. CONCLUSION Fgure 8 α( ) for plane mesh and spatal mesh The modfcaton of the dsplacement response s of %, consderng the dsspaton through frcton. If usng the Raylegh model, the dfference, as percentage, would be of 5%. If coheson nfluence s consdered, the procentual Issue, Volume 4,

8 dfference obtaned s of %. REFERENCES [] Anl K. Chopra, Dynamcs of Structures, Theory and Applcatons to Earthquae Engneerng, Thrd Edton, Pearson Prentce Hall, 7 [] O.C.Zenewcz and R.L. Taylor, Fnte Element Method Vol. The Bass, Ffth Edton, publsher : Butterworth- Henemann Lmted, [] Mrcea Ierema, Slvu Gînju, Numercal analyss of nonlnear structures vol. Computer fundamentals,conspress, Bucharest, 4 [4] Mrcea Ierema, Slvu Gînju, Numercal analyss of nonlnear structures vol. Modellng of structural response,conspress, Bucharest,5 [5] Dan Stematu, Calculaton of hydraulc structures by fnte element method, Techncal Publsher, Bucharest [6] Radu Prscu, Sesmc engneerng of large dams, Techncal Publsher, Bucharest łepeş Onea Florn Date and born locton: 5 june 967, Constantza, Romana; marred,one chld Studes: Techncal Unversty of Cvl Engneerng, Faculty of Hydrotehnc; aprl 999-the publc exponaton of doctorat teze n Hydrotehnc Buldng; Ddactc actvtes of teachng and practcng at the followng subjects: Structural statcs, Strenght of materals, Metal buldngs, Dynamc and stablty; Boos: []łepeş Onea Florn, Structural statcs part II, Ovdus Unversty Press, 4 []łepeş Onea Florn, Structural statcs part I, Ovdus Unversty Press, 6 Issue, Volume 4,