Hydraulic snubbers are extensively used in the nuclear power industry for supporting high energy piping systems subjected to dynamic loadings. These devices allow the piping system to displace freely under slowly applied loads, but lock up under sudden excitations. This paper presents the governing differential equations describing the hydro-mechanical mechanisms of a typical snubber. A finite difference computer code, SNUBER, was developed to solve these equations. Using the code, the load deflection characteristics of the unit were developed for a range of parameters of interest. The parameters included leakage orifice area, initial piston location, eyebolt clearance and reservoir pressures. The results include the load deflection characteristics for various combinations of the controlling parameters and some chamber pressure time history profiles. It is intended that the nonlinear characteristic of the snubbers be incorporated into a structural dynamic analysis program to allow prediction of the overall response of nuclear piping supported by these devices and subjected to a variety of loadings

Curreri, J., Bezler, P.

Hartzman, M.

Subudhi, M.

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— » « , « » THIS VSn ME tUEMtli. -MDHHW4546It hat been reproduce* from the test w l / r _available copy to permit the broadest c <-jssalMe availability.BKL-HUREG—3 454 6DE84 010281Parametric Studies on the Load-DeflectionCharacteristics of Hydraulic SnubbersM. Subudhi, J . Curreri, P. Bezler and M. Hartzman*Structural Analysis DivisionDepartment of Nuclear EnergyBuilding 129Brookhaven National LaboratoryUpton, NY 11973USADISCLAIMERThis report was prepared as an account of work sponsored by an agency of the United StatesGovernment. Neither the United States Government nor any agency thereof, nor any of theiremployees, makes any warranty, express or implied, or assumes any legal liability or responsi-bility for the accuracy, completeness, or usefulness of any information, apparatus, product, orprocess disclosed, or represents that its use would not infringe privately owned rights. Refer-ence herein to any specific commercial product, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom-mendation, or favoring by the United States Government or any agency thereof. The viewsand opinions of authors expressed herein do not necessarily state or reflect those of theUnited States Government or any agency thereof.*U. S. Nuclear Regulatory Commission, Washington, DC.This work was performed under the auspices of the U.S. Nuclear RegulatoryCommission, Washington, DC.- '•-• I.I, ,f *ABSTRACTHydraulic snubbers are extens ive ly used In the nuclear power Industry forsupporting high energy piping systems subjected Co dynamic loadings . These de -v i c e s allow the piping system to displace f ree ly under slotrly applied load*, butlock up under sudden e x c i t a t i o n s . This paper presents the governing d i f f e r e n -t i a l equations describing the hydro-mechanical mechanisms of a typtcal snubber.A f i n i t e di f ference computer code, "SNUBER", was developed to solve these equa-t i o n s . Using the code, the load def lec t ion charac ter i s t i c s of the unit weredeveloped for a range of parameters of In teres t . The parameters Included leak-age o r i f i c e area, i n i t i a l piston locat ion , eyebalt clearance and reservoir pres-sures . The r e s u l t s include the load def lect ion charac ter i s t i c s for various com-binations of the contro l l ing parameters and some chamber pressure time historyp r o f i l e s . I t 1* Intended that the non-linear character i s t i c of the snubbers beincorporated into a structural dynamic analys is program to allow prediction ofthe overal l response of nuclear piping supported by these devices and subjectedto a var ie ty of loadings .NOMENCIJOTREtPBVFAqrADCDP*oDRuDpCrHQ- tineM pressure• effective bulk modulusM lnstsntaneous chamber volume• volumetric flow- piston area- relative displacement of the ends• duct area" flow coefficient- fluid density- total orifice area- diameter of reservoir duct- absolute viscosity- piston outside diameter- clearance between piston and bore- length- force resultant_• .'I I • IL, - total snubber lengthH o - Initial pliton positionB L » fluid bulk modulusBc - container bulk aodulusD « bore diameterD o m outer vessel diameterE - vessel aodulus of elasticityv - vessel Polssun* ratioCfi - Initial clearance between piston and bor*Ai - area of orifice which remains openAj - area of orifice which closesq - end displacementSubscripts:TCRPJiJrm tension- compression• reservoir• piston- joint- i-th end- J-th end" relativeINTRODUCTIONFor years snubbers have been used In nuclear power planta to limit the ad-verse response of piping and equipment produced by dynamic loads. Snubbers aregenerally referred to as dhock arrestors and shock suppressor*. They are attach-ed to the piping with their major axis In the direction of the maximum thermalgrowth and dynamic force. During normal thermal cycles these devices extend andretract with the pipe and thus Induce very little reactive force on the systea.When the pipe is subjected to an earthquake, pipe break, water hammer or reliefvalve actuation etc., these devices lock up and llalt the displacement response^There are two types of snubbers currently available: hydraulic or mechani-cal. Hydraulic snubbers function on the principle of the dashpot where pistonmovement in a hydraulic cylinder is controlled by the flow of fluid through asmall orifice. The higher the force the faster the fluid passes through theopening. Since the orifice size limits the flow of the fluid, the piston rodmovement is controlled by the orifice size. Other arrangement* associated withthe snubber design Improve the dashpot effect and better control the aotlon ofthe piston rod. Due to the compressability of the fluid, these devices absorbenergy under dynamic loading. One problem associated with hydraulic snubbers Isthat they leak and consequentially require constant monitoring and refilling.Mechanical snubbers were designed to circumvent this problem. They have differ-ent operating characteristics. However, they are not examined In this report.This paper presents a mathematical formulation describing the fluid flow Ina hydraulic snubber. The flow through valves, orifices, connecting ducts etc.are considered In the study. The chamber pressures In both the tension and com-pression aide of the piston are determined by the numerical Integration of thecontrolling fluid flow equations. These pressures are then used to calculatethe resultant forces on the piston which is In turn further transmitted to thepiping system. Parametric studies were conducted to determine the effect ofleakage orifice area. Initial piston position and eyebolt clearance on the loaddeflection characteristics of this snubber. In addition cases Involving varia-tions In reservoir pressure were considered.It has been found that the behavior of the hydraulic snubber Is highly non-linear and Is governed by several factora acting simultaneously. The nonllnearl-ty can be expressed by a general bi-llnear curve. The tension and compressionchamber pressures govern the slopes of these curves. In addition, a dependenceon the static equilibrium position of the -piston has also been noted. The... J I.leakage orifice »lie wai found to govern the amount of energy dissipated In onecycle of snubber operation. The larger the ilze, the sore energy dissipated bythe device. Laitly, eyebolt clearance* Introduce a gap-eleaent In the force-displacement characteristic* about the atatlc equilibrium poaltlon.MATHEMATICAL FORMULATIOHSThe arrangement of a typical hydraulic anubber la chovn In Figure 1 where apiston novel within a clcned cylinder filled with a viacoua fluid. The chambersof the bore are connected to a pressurized accumulator or reservoir throughvariable valve and orifices. The valves are designed such that they remain openIf the pressure In the chamber remains belou a prescribed value. When this pres-sure Is exceeded, the valves close and the orifice permits a snail but finiteamount of leakage to ths reservoir and the snubber becomes essentially a hydraul-ic spring. The analysis then Is similar to that of hydraulic valvec found Inhydromechanical control systems [1].The analysis of the system Is based upon the continuity equation and thefluid equation of state. Internal fluid lnertlal effects are excluded. Addition-al equations expressing the fluid flow through the orifices and ducta are used.These are based on laminar and turbulent flow through closed ducts. The develop-ment of the equations was first attempted by Hartzman 12] in a technical note atNRC. In that study a generic snubber design was considered. In the presentstudy an actual snubber design was considered and the following equations u«redeveloped accordingly. The compressibility effects In the chambers are describedby the following differential equations:Tension Chamber: PT - - A f(FT + Fp) - Aj^J U)Compression Chamber: Pc - - B [(Fc - Fp) + A cq r] (2)VCReservoir: P « B [(F_ + F.)] or Constant (3)REquations describing the flow of fluid through the orifice valves and connectingducts:Flow through tension chamber and orifice valves:^ 2 .2 .2 .2 2^JIT)T TIT HD aOTLODFlow from valve to junction, turbulent flow:L.I3 ( J ).25 .75,, -4.75where fj - .2*2 lj uFlow through compression chamber and orifice valves:CDC J (6)uherc"HD "0C"ODFlow from valve to Junction, turbulent flow:r c - fP z " F j ) TJs . (7>cwhere f,, - .242 L_ u ' 2 5 0* 7 5 D "*"75c c cDuct from tension-compression junction to reservoir, turbulent flow:(8)where fR - .242Piston leakage between tension 2nd compression chamber*ra> c3.FP ' 12uH *PT " PC^ * 9 'PThe volume of the contained fluid Is affected by the "loads acting on It.These loads are applied to the fluid ttirough the notion of the piston within thecylinder. The expansion of the cylindrical structure acts In parallel with thevolume changes associated with the bulk modulus of the fluid. This results Inan effective bulk modulus which can be expressed as:B BL Bcwhere B for a thick walled cylinder,I . I (1 + v) D2 + (1 - v) D2.B E °The Instantaneous tension and compression chamber volume and chamber length arecalculated as:(11)mndVC " AC HCHT- L. "(Ho + V (12)HC " Ho + twhere qr - qj - q iThe frictlonal force can be evaluated In terms of a viscous drag forcewhich depend* upon a clearance dimension between the cylinder and piston. Underpressurization the clearance itself changes because of the expansion of thecylinder. To Include thl* effect, the friction force acting on the piston isexpressed as:(13)and F - average pressure acting within the cylinder.The resultant force rate acting on the piston la:Q - V T ~ V c <">From equilibrium of the overall system, the force rates acting at the endswhich Join the structvire are:Qj - -Q. . (15)The total force acting on the snuhber at any time Is govern as:Qj " 'I Qjdt + QfThe characteristics of the snubber are primarily controlled by the orificevalves. Each orifice valve has two orifice openings: one which closes when thepressure difference across the valve exceeds a predetermined valve (F r l t ) . andone which remains open to provide pressure release in the loaded position. Thearea of each orifice is therefore defined as follows:Tension Orifice:where Bj » 1 if (P, - P,) <_ F . O7>- o if <p, - P j ) > p c r l tCompression Orifice:*VC " *1C + BC*2Cwhere Bc - 1 If (Pj - Pj) < P c r i t (")" 0 If (P, - F ) > PHETHOD OF SOLUTIONThe computer code SNUBER processes the above equations. The code currentlyuses a s inusoidal motion with spec i f i ed amplitude and frequency as Input. How-ever, any defined displacement time history could be used as the Input forcingfunction. Xn addit ion, the code w i l l accept an Input-aotlon that i s appliedthrough a clearance gap. Thla l a acconplished In the code through an engage ordisengage comand that recognizes when the clearance gap has been transversed.The force-displacement response of snubber I s obtained by numerical Integra-t ion of the above equations. The program uses the Newtoii-Raphson numericalGcheme to i t e r a t e the equations simultaneously while accounting for the s t a t e ofthe o r i f i c e va lves and the end condit ions . This y i e l d s the flows snd the5 u/':(.'and the Incremental pressures and hence, the total Internet pressures acting onthe piston at any Inatant, I.e., for any displacement of the platon relative tothe anubber body. The total force acting on the anubbcr la then obtained byapplying the equation* of equilibrium.RESULTSAnalyaea were performed for varloua snubber dealgna. Computer runa forvarloua cases have been obtained. These Include caaea where one of the desirnparameters waa varied from the original or baseline configuration of a typicalsnubber. The parameters varied In this study were:(1) Leakage orifice area.(2) Initial equilibrium piston position(3) Input forcing frequency(A) Variable reservoir pressure(5) Direction of load application(6) Eyebolt clearanceFor the baseline configuration the values of these parameters were:Leakage orifice area - 0.5X of valve areaInitial equilibrium piston (not In center) - 3.625 In. (9.21 cm)Input forcing frequency • 30 HzConstant reservoir pressure - 20 palg (137.9 x 10 K/a )Direction of load application - compression chamber init ial direction.£yebolt clearance * 0.The force-deflection characteristics for each caae are shown in flgurea 2through 8. Figure 2 depicts the characteristics for the baseline configurationvhile the remaining figures depict the characteristics when one or another para-meter are varied.Referring to Figure 2, the following observations, which are conaon to allcases except where noted, can be made. The Initial strolce is In the positivedirection causing compression In the compression chamber. This motion Is resist-ed with high apparent stiffness, initial slope of curve. On the return stroke ahysteretic loop is formed indicating energy dissipation within the device, thegreater the loop the greater the energy dissipation. After the null position lareached compression of the tension chamber ensues with the motion being resistedwith a lower apparent stiffness. The observed initial difference In stiffnessto notion In the positive or negative directions is due to the difference In borelength of the compression and tension chambers associated with the Initial offcenter piston position assumed for the baseline configuration. With repeatedcycles a shift towarda a steady state characteristic la observed.Figure 3 presents the reaulta for the caae where the leakage orifice area 1*Increased to 0.014 in. (0.09 CM 2). The dominant effect of this change is tomarkedly Increase the size of the hyateretle loop. The energy diaalpated by thedevice Is apparently directly proportioned to the leakage orifice area. Somechanges In apparent stiffness and maxlauim force can also be noted.Figure 4 shows the force-displacement characteristics when the Initial posi-tion of the piston Is aligned at the center of the cylinder initial piston posi-tion - 3.25 In. (8.26 cm). The response curve* are very different from the otherresults. The slope and the maximum load are significantly altered becoming sym-metric. This ateady atate characteristic could be represented by a linear spring.Figure 5 corresponds Co the caae where the input forcing frequency Is 15 HzInstead of 30 Hz. With the slower Input the device dissipate* a greater amountof energy. An increase In the stiffness of the device, both In tension and com-pression la alao observed.Figure 6 shows the response for th t cas t where the reservoir volume 1*taken email enough to s i g n i f i c a n t l y vary chwber pressure with the flow of f lu idInto and out of the chaabcr. The pressure changes are governed by the d i f f e r e n -t i a l Equation ( 3 ) . The s lope and the maximum load va lves for t h i s c s s e shows i g n i f i c a n t changes. Unlike the other c a s e s , these curves are symmetrical aboutthe zero displacement point even though the I n i t i a l p o s i t i o n of the p i s ton I s notat the center of the c y l i n d e r . For t h i s configuration the device could be repre-sented with a b i - l i n e a r spring.Figure 7 shows the r e s u l t s for the case where only the d i rec t i on of the I n i -t i a l input displacement I s reversed. This a l lows the tension s ide of the p i s tont o a c t i v a t e f l r a t . Unlike the conclusions by Hartzaan [ 2 ] , the general charac-t e r i s t i c s of t h i s case remain unchanged when compared with Figure 2 . Only s l i g h tchanges In the c h a r a c t e r i s t i c curves are evident .Figure 6 I l l u s t r a t e s the l a s t case where an eyebolt clearance a f f e c t l a con-s idered . An eyebolt clearance of .0015 In . (.0038 ca) was assumed t o e x i s t a tthe pipe and snubber attachment. In t h i s case , the response e x h i b i t s a deadzone a t the center when the piston i s In the clearance range. During t h i s periodthe pressur* In the chambers remain constant . The o v e r a l l s lopes of each aideof the p l o t s remain almost unchanged.Figure 9 and Figure 10 show t y p i c a l t races of the f low Into the reservoir(Fg) and the Junction pressure (Pj) for the case corresponding t o Figure 3 . I ti s noted that a f t er an i n i t i a l t l a e the flow and the pressure converge t o asteady s t a t e pattern under the c y c l i c loading. The spikes on these p l o t s - c o r r e -spond to the t imes when the o r i f i c e v a l v e s open or c l o s e under the f low and pres -sure condi t ions .CONCLUSIONSIn aosC cases the response c h a r a c t e r i s t i c s exhibited by the hydraulic snub-ber investigated was a bl-linear curve. The blllnearlty Increase* as the staticequilibrium position of the piston moves away froa the center location of thecylinder. This Is due to the compressibility effect of fluid In each chamberwithin the cylinder. Since this factor defines the stiffness of spring actionby the fluid and also depends on the volume of liquid behind each piston aide,the bilinear Ity becomes very distinct when the Initial platon position approachesthe ends of the cylinder bore. When the Initial piston position Is near the cen-ter of the bore, the bilinear characteristic converges to a mono-linearcharacteristic.Leakage through the leakage orifice and leakage past the piston have a s la i -lar effect on the characteristics. They both Increase the dissipation energy andhence the damping in the system. A similar effect was alao noted for the lowerinput frequency. The higher damping also increases the number of cycles neces-sary to attain a steady-state characteristic. The overall characteristic of thesystem however, remains unchanged whether the snubber Is Initially subjected toa push or pull loading.Eyebolt clearance In the syatea Introduces a dead zone In the force-displacement characteristic about the static equilibrium position. This strictlyIntroduces acre nonllnearlty Into the anubber behavior, which will be reflectedin the overall piping response. Finally, the reservoir pressure condition playsan Important role In the anubber response. The snubber acta bl-llnear or aono-llnear depending on the Initial piston position and steady state behavior i sreached after a alnlaua number of cycles.It can be concluded that hydraulic snubber behavior depends on a nuaber ofparaaetera. The simple assumption of elastic behavior used In current pipinganalysis Is not Justifiable. However, In a l l caaea the snubber characteristiccould be approximated with a bl-llnear spring aodel. Further studies with earth-quake loadinga and sudden valve closure loadinga should be made In order tounderstand the actual behavior of the device.REFERENCES1. Merrlt, H. E . , "Hydraulic Control Systea", J . Wiley and Som, Inc.Mew York, 1967.2. Hartzaan, M., "Load-Deflection Characteristic* of Hydraulic SnubberSupport, USNRC-MEB Technical Note.(FIXED)FIG. 1: SCHEMATIC DIAGRAM OF A HYDRAULIC SNUBBERX(2.54 X 10" ) Displacement in Cits.FIG. 2: FORCE-OISPLACEMENT CHARACTERISTICS FOR AS-DESIGHED DATAPi-1X(Z.54 X 10"') Displacement in Cfls.FIG. 3: FORCE-DISPLACEMENT CHARACTERISTICS FOR ORIFICE PARAMETEREQUAL T0-30J OF VALVE AREAX(2.54 X 10"1) Displacement 1n Cfls.FIG. 4: FORCE-DISPLACEMENT CHARACTERISTICS FOR INITIAL PISTON POSITIONAT CENTERC'-\>VX(2.54 X 10"') Displacement in CMs.FIG. 5: FORCE-DISPLACEMENT CHARACTERISTICS FOR A FORCING FREQUENCYf = 15 HzFIG.X(2.54 X 10'1) Displacement 1n Clis.6: FORCE-DISPLACEMENT CHARACTERISTICS FOR VARIABLE RESERVOIR PRESSURE(hiX(2.54 X 10"1) Displacement in CMs.FIG. 7: FORCE-DISPLACEMENT CHARACTERISTICS FOR A REVERSE INITIAL MOVEMENTX(2.54 X 10" ) Displacement 1n CMs.FIG. 8: DISPLACEMENT CHARACTERISTICS WITH AN EYE-BOLT GAPS ?ATime in sees.FIG. 9: RESERVOIR FLOW CHARACTERISTICS FOR A TYPICAL SNUBBERI AS IN ("G- 3)\iTine in sees.FIG. 10: PRESSURE CHARACTERISTIC AT THE JUNCTION FOR A TOPICAL'. ^SNUBBER AS IN (FIG. 3)

Parametric studies on the load-deflection characteristics of hydraulic snubbers

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Parametric studies on the load-deflection characteristics of hydraulic snubbers

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