Different Approaches for the Modelling of High Viscous Dampers in Piping Dynamic Analysis. Acceptable Limits for Simplifications.

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1 0th Internatonal Conference on Structural Mechancs n Reactor Technology (SMRT 0) Espoo, Fnland, August 9-4, 009 SMRT 0-Dvson V, Paper 833 Dfferent Approaches for the Modellng of Hgh Vscous Dampers n Ppng Dynamc Analyss. Acceptable Lmts for Smplfcatons. Alexey Berkovsky a, Peter Vaslyev a and Oleg Kreev a a CKTI Vbrosesm, Sant Petersburg, Russa, e-mal: bam@cvs.spb.su Keywords: Hgh Vscous Damper, Modellng, Maxwell Model, Dynamc Analyss. ABSTRACT Durng recent decades Hgh Vscous Dampers (HVD) have been ntensvely mplemented n Nuclear and Conventonal Industry for protecton of ppng systems and equpment from the wde range of dynamc loads: earthquake, water/steam hammer, operatonal vbraton, etc. Applcaton of these devces requres from the Desgner/Analyst to mplement a proper procedure coverng all stages of desgn: selecton of damper's locaton along ppelne, choosng damper's type, and fnally modellng of damper n the frame of ppng dynamc analyss. Presented paper addresses namely the last ssue: modellng of vscous dampers n ppng sesmc analyss. It s well-known that Hgh Vscous Damper exhbts essental frequency-dependent characterstc of dynamc stffness that hardly could be descrbed by the conventonal approach avalable n most commercal ppng software programs: representng of damper's acton by a sprng element actve for dynamc loads only. From the other hand more sophstcated 4-parametrcal Maxwell model (Kostarev et al, 993) that allows accurately reproduce damper's characterstcs over frequency range of nterest s not wdely used n the specalzed ppng software. Besdes, applcaton of Maxwell model requres performng of Tme Hstory Analyss whle the conventonal desgn procedure for sesmc calculatons s Response Spectrum Method that uses Floor Response Spectra as sesmc nput. Paper presents several numercal examples of ppng calculatons wth dfferent models of dampers and dscusses acceptablty and lmts for mplementaton of a smplfed approach when damper s modelled by means of sprng elements. INTRODUCTION Vscoelastc Ppework dampers or wth another name of Hgh Vscous Dampers (HVD) have a long and successful story of mplementaton for sesmc upgradng and vbraton control of ppng systems and components for dfferent nstallatons: NPPs, conventonal power plants, ndustral facltes. As a dynamc restrant HVD s a devce that works n a softer manner than snubbers provdng to the system essental addtonal dampng. Hgh dampng n the devce s a result of deformaton of an extremely hgh vscous lqud that s located n the space between damper s pston and housng, Fgure. Sgnfcant pecularty of HVD s nonlnear dampng and stffness characterstcs aganst frequency of exctaton. It was shown n a number of works (Kostarev et al, 993; Berkovsky at al, 995, Lewandowsk at al, 007) that such dependence could be satsfactory approxmated by four-parametrcal Maxwell model. For purposes of ppng dynamc analyss ths model could be easly ntroduced n a frame of Tme Hstory Analyss by means of sprng and deal damper elements connected n lne. However the dffcultes exst for mplementaton of such model n the frame of conventonal Response Spectrum Method (RSM) that realzed n most commercally avalable ppng software and has been stll n use as a man desgn tool for sesmc analyss. So, the ntent of ths paper s to assess the possblty and acceptable lmts for mplementaton of HVD smplfed model by means of sprng elements n the frame of conventonal RSM, to defne accuracy of such Copyrght 009 by SMIRT 0

2 an approach and to develop recommendatons for software upgradng for a correct modellng of HVDs, f necessary. Fgure. Typcal Vscoelastc Ppework Damper: outsde vew and cross-secton (courtesy of GERB Schwngungssolerungen GmbH & Co. KG, Berln) 3 MATHEMATICAL MODEL OF HIGH VISCOUS DAMPER As t follows from the numerous test results and Manufacture Catalogue [GERB] dynamc stffness of HVD s sgnfcantly changed over frequency range, snce the damper's dynamc propertes are functon of workng lqud as well as arrangement of damper's nternal elements. Workng Vscous Lqud used n dampers defnes damper's vscous-elastc behavour. The smplest mathematcal model descrbng such behavour s a Maxwell Model whch conssts of connected n lne deal vscous damper and sprng element, Fgure. K B Fgure. Maxwell Vscous-Elastc Model Such model demonstrates the followng features, whch are typcal for the measured HVD propertes: the reacton of HVD at the low frequency loadng range s consdered as a vscous and may be descrbed by an expresson: R = -B*v, where R reacton force, v velocty of a pston relatvely to the housng, B dampng resstance; for the hgh frequency loadng range the damper's reacton shows essentally elastc character and may be descrbed as: R = -K*x, where x relatve dsplacement "pston-hosng", K stffness rato. In case of the Maxwell Model, K s a stffness of sprng element, B dampng coeffcent for the deal vscous element. The parameter named as a characterstc frequency of Maxwell model s often used n varous applcatons s ω 0 = K/B. Copyrght 009 by SMIRT 0

3 In case of damper's harmonc loadng, when the damper's pston s moved relatvely the housng wth frequency ω (x = x 0 *sn(ω*t)), the damper's reacton force would be harmonc as well, but shfted on the some phase angle: R = x 0 *C e *sn(ω*t) + x 0 *C v *cos(ω*t), where: x 0 oscllaton ampltude, ω - angular frequency of forced oscllatons, t tme, C e elastc part of damper's dynamc stffness, C v vscous part of damper's dynamc stffness. Damper's reacton force may also be rewrtten n a form: R = x 0 *C s *sn(ω*t + ϕ) () where: / C s = (C e + Cv ) equvalent damper's stffness for a gven frequency, ϕ phase angle (tg(ϕ) = C v /C e ). It should be noted that damper's stffness components are hghly dependant on the frequency ω. These dependences for Maxwell Model have the followng form: C e = K*(ω/ω 0 ) /( + (ω/ω 0 ) ) C v = K*(ω/ω 0 )/( + (ω/ω 0 ) ) () Graphcally they are shown n Fgure 3. Fgure 3. Dependence of Maxwell Model Characterstcs from the frequency. The frequency rato shown n ths plot s ω/ω 0, stffness rato C/K. When the frequency of vbraton acheves the value of characterstc frequency ω 0 elastc and vscous stffness parts become equal and make one half from the sprng stffness value K. In the frequency range less than ω 0 the vscous part s domnated and contra versa: for the hgh frequency range the elastc component s prevaled. Change of the phase angle ϕ dependng from the frequency s shown n Fgure 4: 3 Copyrght 009 by SMIRT 0

4 Fgure 4. Dependence of HVD phase angle from the frequency rato. It should be noted that dynamc characterstcs of the real Hgh Vscous Dampers derved from the experments are more complex than those shown n Fgure 3. However, t was recognzed that a set of two Maxwell chans demonstrates qute approprate results for engneerng purposes. Such model s presented n Fgure 5: Fgure 5. Scheme of the mathematcal model of Hgh Vscous Damper. Under such approach the expressons () are transformed to the followng form: C e = K *(ω/ω ) /( + (ω/ω ) ) + K *(ω/ω ) /( + (ω/ω ) ) C v = K *(ω/ω )/( + (ω/ω ) ) + K *(ω/ω )/( + (ω/ω ) ) (3) where: ω = K /B and ω = K /B characterstc frequences for frst and second Maxwell chans Thus, the problem of creaton of the mathematcal model for Hgh Vscous Damper based on the expermental data s reduced to defnton of four unknown parameters of model. Lets assume, that N sets of expermental data n form of {ω, Ce, Cv, =,,.. N} are known. Then, the target functon can be wrtten down as the sum of squares of resduals between the expermental and «theoretcal» values of stffness components: S = Σ((Ce Ce(ω )) + (Cv Cv(ω )) ) (4) Unknown parameters of model are defned then by fndng of mnmum of target functon by the use of any of optmzaton methods. It should be noted that the classcal least-squares method n ths case does not work snce the model's parameters ncluded n equatons are non-lnear. The Fgure 6 shows an example of expermental data approxmaton for determnng of HVD model characterstcs: 4 Copyrght 009 by SMIRT 0

5 approxmaton expermental data Fgure 6. Approxmaton of the Expermental Data wth the use of 4-parameters Maxwell Model of HVD 4 SIMPLIFIED MODEL OF HVD Descrbed above modelng of HVD by means of 4-parametrcal Maxwell model s sutable for realzaton n the frame of the Tme Hstory Analyss (THA). However, for desgn purposes such method of sesmc analyss s used rather rarely. Moreover, Maxwell model of HVD s realzed only n a few software packages: ROHR ( and dpipe ( Most commercally avalable ppng software packages stll use for dynamc restrants only "snubber" model: sprng element that does not resst statc loads and s actve only for a dynamc mpact. The benefts of such modelng s a possblty to mplement conventonal response spectrum (RSM) method for a soluton. From the other hand, t s clear that such approach neglects dampng ntroduced by HVD n analyzed ppng system that s a core feature of HVDs and postvely nfluences on dynamc capacty of a system. The followng dscusson s devoted to results of ppng sesmc analyss when HVD s modeled by means of "snubber" sprng element wth equvalent stffness chosen on the bass of some "character" frequency that defnes damper's response under certan exctaton. One of such procedures that could be mplemented to defne equvalent damper's sprng rato s shown n Fgure 7. Use of ths algorthm nvolves teratve calculatons of ppng natural frequences and mode shapes up to convergence of character frequency for each damper. 5 Copyrght 009 by SMIRT 0

6 6 Copyrght 009 by SMIRT 0 { } ( ) T d k K K ω Φ + Φ = n k char n K F F K φ φ = = = k k char char k char F F R F = max R ε < Fgure 7. Flow-chart for teratve procedure of selecton damper's "character frequency"

7 To check acceptablty of such approach several ppng systems from the real applcatons were taken for numercal evaluaton, Fgure 8. HVD of dfferent types were nstalled on consdered ppng to reduce sesmc loads. These systems chosen from the ndustral and nuclear ppng are dfferent n szes, layout and modal propertes, Table. For THA sesmc loads appled to analyzed systems were defned n terms of three-component accelerograms. Response spectra corresponded to these accelerograms are also dfferent n shape and frequency content: from the broadened desgn spectra to rather narrow spectra wth sharp peaks on some frequences. To nvestgate dfference between two approaches: Maxwell model + THA vs. Sprng Model + RSM the followng procedure has been mplemented:. Baselne soluton was taken from THA wth representaton of HVD my means 4-parametrcal Maxwell model for each spatal drecton. Sesmc nput was defned n terms of threecomponent Tme Hstory Acceleraton.. Acceleraton Response Spectra used as Sesmc nput n frame of Response Spectrum Method analyss were generated drectly from the gven TH acceleraton records wth a fne frequency step (0. Hz) and dampng rato consstent wth those used n THA. 3. In the frame of RSM each HVD was modeled by means of three-component sprng elements wth sprng rato defned accordng to the procedure descrbed above. 4. For comparson purposes procedure descrbed n the Ppng Benchmark Problems (NUREG) was accepted: a. each consdered ppng model was dvded on several segments (from fve to nne, dependng on ppng length); b. comparson of maxmum values of each component of ppe dsplacements and maxmum SRSS values of moments has been performed for each segment ; c. reactons of all ppng supports were compared as well on the component to component bass. 5. Each compared parameter was expressed as rato between RSM and THA. Thus, values hgher than.0 correspond to a level of conservatsm of appled method and vce versus: values less than unty ndcate degree of non-conservatve estmaton. 6. For statstcal processng of results the followng values were plotted: maxmum, mnmum, mean and mean value mnus one sgma (standard devaton). These values were obtaned by processng correspondng results for all ppng segments of each consdered model. Results of these comparatve analyses are presented n the form of stock dagrams n Fgures 9. Table. Descrpton and propertes of analyzed ppng. Ppng Range of natural Descrpton Model frequences ( up to 33 Hz) ) (FW) Conventonal Power Plant Feed Water Lne 43 natural frequences from DN00 DN50.5 Hz (HPP) Conventonal Power Plant Hgh Pressure 4 natural frequences from and Temperature Steam lne (from DN Hz DN400) 3 (IS) Industral Ppng (DN400 DN800) 58 natural frequences from.94 Hz 4 (JND) Nuclear Safety Related Ppng (DN50 93 natural frequences from DN300) 0.85 Hz 5 (KO) Nuclear Class Ppng (Pressurzer system), 40 natural frequences from DN Hz ) frequences are gven for unrestraned ppng Number and type of nstalled HVD 3xVD-35/9-7 4 HVD: from VD-59/76-7 to VD-46/9-5 7 HVD: from VD-35/9-7 to VD-630/46-5 3xVD-35/9-7 + xvd-46/35-7 xvd-9/ Copyrght 009 by SMIRT 0

8 Model (FW) Model (HPP) Model 3 (IS) Model 4 (JND) Fgure 8. Ppng test models. Model 5 (KO) 8 Copyrght 009 by SMIRT 0

9 3.5 mean max mn mean-sgma Rato = RSM / THA FW HPP IS JND KO Ppng Models Fgure 9. Statstcal processng of analyss results. Dsplacements.5 mean max mn mean-sgma Rato = RSM / THA FW HPP IS JND KO Ppng Models Fgure0. Statstcal processng of analyss results. Moments. 9 Copyrght 009 by SMIRT 0

10 mean max mn mean-sgma Rato = RSM / THA FW HPP IS JND KO Ppng Models Fgure. Statstcal processng of analyss results. Support's reactons 5 CONCLUSIONS. Accordng to obtaned results the followng conclusons can be drawn: a smplfed model of HVD generally leads to an overestmaton of the sesmc response of the system. The smplfed model of HVD provdes n general conservatsm n calculaton of ppng system dsplacements and element's nternal moments (stresses). However, the values of support's reactons n some cases are essentally underestmated, that can not be accepted for the purposes of desgn. Thus, t seems that up to now the only one relable method for sesmc desgn of ppng systems equpped wth HVD s the Tme Hstory Analyss. Nevertheless, the smplfed method may be useful on the ntal stages of desgn when t s necessary to locate dampers along ppelne to reduce sesmc stresses.. It seems that further studes are needed to develop relable applcaton of the Response Spectrum Method for ppng systems wth non-proportonal dampng. For the accurate mplementaton, such approaches should address several mportant ssues: a. summaton rules for mode shapes wth hgh dampng rato (hgher than 30 %) b. treatng of overdamped modes (dampng hgher than crtcal) c. forms of nput floor response spectra (evdently that besdes conventonal acceleraton response spectra, response spectra defned n terms of velocty wll be needed). 3. Some of exstng conventonal software for a ppng analyss probably wll need some upgradng for an accurate modelng of HVDs consderng that snce 008 ths devce s a standard dynamc support accordng to ASME BVPC Secton III Subsecton NF (supports). 0 Copyrght 009 by SMIRT 0

11 REFERENCES Kostarev V., Berkovsky A., Kreev O., Vaslev P., Applcaton of Mathematcal Model of Hgh Vscoelastc Damper n Dynamc Analyss of NPP Ppng and Equpment, SMIRT-, 993, paper K0/6 Berkovsky A., Kostarev V., Schukn A., Vaslyev P., (995), Sesmc Analyss of VVER NPP prmary coolant loop wth dfferent asesmc devces. Transactons of the 3th SMRT, Porto Alegre. Berkovsky A., Kostarev V., Schukn A., Kostov M., Boadjev A., Boadjev Z., (997), Sesmc analyss of the safety related ppng and PCLS of the WWER-440 NPP. Transactons of the 4th SMRT, Lyon. J. Novotný, J. Novotný jun., Response of the Steam Generator VVER 000 to a Steam Lne Break, Transactons of the 7th SMIRT, Paper # J06-, Prague, 003 Kostarev V., Berkovsk A., Schukn A., Upgradng of dynamc relablty and lfe extenson of ppng by means of hgh vscous damper technology. Transactons of PVP ASME Conference, Boston 999 NUREG/CR-6049, Ppng Benchmark Problems for the General Electrc Advanced Bolng Reactor, 993 R. Lewandowsk, B. Chorążyczewsk, Remarks on Modellng of Passve Vscoelastc Dampers, 9 th Internatonal Conference "Modern Buldng Materals, Structures and Technques", Vlnus, 007 Ppework Dampers, Techncal Report, GERB Schwngungssolerungen GmbH & Co. KG, 997 Copyrght 009 by SMIRT 0