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Tainter gates installed in the Folsom dam in California have a circular-arc skinplate with a height of 15.5 m and a radius of. 14.33 m .... The 0.7 mm rectangle ... for the Folsom Dam Tainter-gate, shown in Figure 5. .... reduced rotation center height (r:=Rs/do) ... about 13 mm the 3-D equivalent model gate struck the channel.
143

143

P『 oceedings of PVP2006・ iCPV「 Ⅱ l l 2006 ASME Pressu『 e Vesseis and Piping Division Conference

Ju:y23Ⅲ 27,2006,Vancouver,BC,Canada

PVP2006-ICPVTll-93917

ViBRAT:ON TESTS W:TH Al′ 13・ SCALED 3‐ D MODEL OF THE FOLSOM DAM TA:NTER‐ GATE AND:TS PREDICT:ON BY THEORY

Kelko ANAMI Dept.of Mechanical Engineering,

NoriakiiSH‖ Dept.of Mechanical Engineering,

Ashikaga lnstitute of Technology

Osaka Electro‐ Communicalon Universty

Ashikaga,Tochigi 326-8558,Japan anamiα )ashlech.acjp

Neyagawa1 0saka 572‐ 8530,Japan ishil(Disc.OSakac.ac.jp

Char:es W.KN:SELY

Tatsuya OKU

Dept.of Mechanical Engineering,

Dept.of Mechanical Engineering,

Buckne‖ University

Osaka Electro¨ Communicalon Univ. Neyagawa,Osaka 572‐ 8530,」 apan

Lewisburg,PA 17837,USA knisely(DbuCkne‖ edu

d04201(DiSC.OSakaC.ac.jp

ABSTRACT This study prescnts 3-D modcl gatc vibration test resul偲 delnollstrating vi01ent spontallcous vibrations and validating the

basic assumptions made in prevlously published ■eoretical analyses. First, thc design of a 1/13-scaled 3-lD model of Folsom dam Tainter― gate is presentcd,m which the streamwise ︵ 3︶日 つヽ曽

natural bending vibratiOn mode of the skinplate,measured m the fcld vibration tcsts on the remainhg Folsom gatc,is shown to be correctly replicated wi■ the aid of FEM shulations. SecondlyЪ in― att and in― 、 vatcr vibration test results 、 vith thc 1/13-scaled 3¨

E)model arc prcscntcd,rcproduchg伍 c intcnsc

coupled¨ modc sclf― cxcitcd vibratiOns.Thirdltt tcst rcsults arc

典 tted On a theoreticJly calcuhted stablity critc五

on diagram

cOnflHn thc validity of tllc theoretlcal analysis.Finany,thc

lntcnse dynamic ulstability of thc Folsom gatc,which could

Figure l: Side view of an 87-ton Tainter gate from the

havc caused its failurc,is prcscntcd.

Folsom Dam in Calrornia, showing two predominant natural vibration modes.

:NTRODUCT10N Tainter gatcs mstalled h the Folsom daln in California have a circular― arc skinplate、 vith a height of 15.5 nl and a radius of

14.331■ ,as shown in Figure l

The spanwise lcngth is 12.8m sized Tain撤 gates failed during operation On July 17, 1995 A similar and thc gross mass is 87 tons Onc ofthe largc―

Tainter gate failure occurred in Japan in July 1967, as documcnted by lshii`′ α′ (1980) A self― excitcd vibradon

mechanism which could have causcd the gatc failure was forllnulated and studied by lshii&Imaichi(1977),Ishii

α αl

(1977)and IShii&Naudaschcr(1984,1992). Immediatcly aner tllc Folsom daln gatc failure, Ish五 (1995a, 1995b)ulveStigatcd the causc of the failure, The rcsults■ om thcsc studies suggested that■ e gate may have

cxpe五cnced a new typc of violent scliexcited vibration that 、 vas couplcd、 vith the n。 、 v rate variation bencath the gatc and that was different from tlle suggested mechanism for the gtte Lilure h Japan.

ln dle Folsom dam failure analysis,expcrimental modal analysis of a geometncaly silnilar Tainter gatc was carncd out primarily tO establish the mttor vibratiOn modcs ofthc gatc[sce

Anami&Ish五 (1998a)].As a rcsult,the strcalnwisc bcndhg vibration modc of thc skinplatc, shown h Figurc 2, was

idcntiflcd.On the spanwisc― radial plane,the skinplate shows a

haliwavelcngtll bendhg mode with the nodes at both the spanwise ends, while on the strealnwiseverical plane, tlle

skinplate exhibits the flrst stleamwise bending mode with the singlc nodc ncar thc sccond radial am(1=4 in Figurc 2),that is,thc gatc undergocs rOtational streamwise vlbration about the

l‐ 1

nodal line.

Vibration in this modc pushcs and dra、 vs the watcr in thc upsttcalm rcservoir, thus illducing an cxtrcmcly largc hydrodyllamic prcssure in thc reservoむ [See Anami,Ishi&

Yamasaki(2000)]Such exccss市 C push_狙 d― draw pressurcs

5

result in a signiflcant added mass effect, which drasticaHy lo、vcrs

6

thc vibration frcqucncy of thc skinplatc from that fbund

7

in air[scc Anami&lshii(1998b)]The COuplhg of this low

8

frcquency streamwise vlbration of tlle skinplate with tlle rotary

bOdy modc about thc trunnion plll can induce a violent excited vibration with tlle aid of the ■ow― ratc vanation

rigid― self‐

undcr the gatc[see Allalni&Ishii(1999,2000,2001,2003)],

Figure 2: Streamwise bending vib「 ation mode of the

alld hcncc could dcstroy thc gate,as dctailed in Allarni(2002) Initially, this dangerous coupled― modc self― cxcited

skinplate of Folsom Dam Tainter gate.

vbration、 vas demonstratcd ofthc Folsom Dam gatc by Ananll

&Ish五 (2003). Furtllcr, thcorctical analyscs havc bccn also undcrtaken by Anami&Ishii(2000)tO Calculate thc lcvel of dyllarnic instability of Folsom Dam gatc at itt failurc,and to ablish a dynamic dcsign criteria for Taintcr gatcs to assurc `ung ternl stable opcration.In tllesc thcoretical analyscs,it was

indamentJly assumcd hat tllis skhplate rotational stream、 visc bcnding vibration could bc sufEciently、 vell representcd by all 、 vith very small cquivalcnt rigid bOdy rotational vibratiOn amplitude ln thc prescnt study. flrst, rnodcl tests 、 vith a largc 1/13scalcd 3-dimcnsional model gate of the ■11lcd Folsonl Dam Tainter―

gate are conducted, dlus ultilnatcly conirlning thc

susccptibillty of thc failed FolsOm Daln Talllter gatc to thc dン namic

hstability leadulg to a vcry violcnt sclfcxcitcd

Figure 3:Back and side views of 1/13-sca!ed 3¨ D exact

model of Folsom dam Tainter gate.

vibratiOn. SccOndly, the measured data 、 veК plottcd on thc vith the stability criterion diagram theorctically calculatcd 、 fundamcntal assumption Of skin platc rOtational vibration,thus validating tllc thcorctical analyscs. Flllanン Ъthe thcorcticaHy calculatcd stability critcriOn diagram for the Folsom gate is prcscntcd,along witll thc datum point ofthc Folsom gate at its incipient failure, thus conflrlning mat the intense dynanlic instability could havc caused the failurc lノ 13Ⅲ

SCALED 3… D:MENSiONAL

Flrst,a1/13-scalcd

EXACT MODEL 3-dhncnsional exact inodel of

多,啜

■m Tahter gate was made with stainless steel Thc back and

sidc vicws are shown in Figurc 3, in which the cLcular skinplatc has a hcight of l.199 nl, a radius of l.108 m and a span、 visc length Of 0 99 m The ccnter ofthe skinplatc cxactly vcrc cOincides with tllc trunnion pin cent(ォ All membcrs 、 rigidly connccted by means of、 vclds fl・ om an clcctric wcldcr Thc do、 vnstrcam― sidc vic、 v ofthc 3-diincnsional inodcl gatc is

shown in Figurc 4

'

Folsom

午デ



Figure 4: Downstream― side view of 1/13-scaled 3¨ D exact model of Fo!som dam Tainter gate_

Ll― ai mOdal analysis (impulsivc

hallllncrhgl testS WCrc conducted on the 3-dimensional model

gtte to dctcrmine the sklllplate strcamwisc in― air natural

vibration icqucncy of about 200 Hz. This is bccause the natural vibratiOn tcqucncy is 13 t缶 nes higher than the prototン pc,if alllengths are reduced gcometrically by a factor of

13 alld thc mOdulus of elasticity remains about thc samc This natural vibration frcquency was too high,but testing lll watcr 、 vas done h casc thc addcd mass ofthe watcr would lo、 vcr it sufflciently for testing purposcs

The exact 3-dimensional model gate was hstaned in a nume with a hcight of l.2 nl,a widul ofl o nl and a lcngth of vatcr natural 8.0■ 1,to conduct in― watcr vibration tests Thc in― 、 vbration tcquency was lowered to about 59 98 Hz duc to tllc vatcr. Howcvcr, this in― watcr added mass effect of thc 、 frcquency was still too high to pcrform a corcct vibration tcst

for thc Folsom Daln Taintcr― gatc, bccausc thc flow_rtte va五 ation undcr thc gatc,bcnlg sub」 cct to tlle gravity fOrce,

Copyright(⊃

2006 by ASME

that is,with 10wcr natural vbration frequcnc"was designed and cOnstructed h a sccond iteration. 1ノ

13Ⅱ SCALED

3… D:MENS10NAL

EQUIVALENT MODEL

Thc indamcntal structurc was modifled to permit hc 1/13-scaled model to maintain an in― 、 vater natural vlbration frequency closc to thc same valuc as for thc protOtype Taillter gate Then,tlle proposed structurc with low streamwisc五 gidity

was analyzed using an FEM simulation.

The techniquc for

this FEM simulation for a complicated structurc like a Taintcr

gate was already cstablished by Allami(2002),Anami,Ishii, Kniscly&Oku(2005)The teCllniquc was applied iterat市 ely to



dcsign a inodcl whidh is a comparativcly flexl)lc structurc but

can stiH reproduce prOtotype vbralon mode correctly. In order to rcproduce in thc modelthc same vibratiOn mOde as tl■

Figure 5:A1/13-scaled 3‐ D equivalent model by FEM.

c Folsom Dam Tainter gate,the static defomation of the

skinplatc due to hydrostatic load needed to be tlle same as tlle

actual deformation.Va五 ous structures wcrc cxamilled bascd on this iundamental vicwpoint Consequentlン Ъit was deterlnincd

1

that a more nexible skinplatc model of tllc structurc was

optimal The FEM model of the 1/13¨ scaled 3-dimcnsional cquivalellt modcl is shOwn in Figurc 5.Thc O.7 nlm rcctanglc stainlcss platc(1199× 990 nlm)of the skinplatc was supported evcry 1 60 mrn in the spanwise dircction with 7 circular― arc rlbs

(CaCh With a thickncss of5 mm and a width of50 mm).It Was supportcd by 8 flat bars each、 vith a 10 1nm tllick and 50 1nm widc crOss― scction connectcd to the trunnion pin with a diamctcr of 30 mm.The structurc is considerably simpler than

thc Folsom Daln Tainter gate[SCC Anami(2002)]

The static dcfomlation(dctermine by FEM)duc tO the hydrostatic watcr load at a dcptll of 1 084 mm(1/13 thc watcr dcptll atthe thc ofthe Folsom Dam g■ c failure)is ShOWn in Figurc 6.Tllc bOttom centcr of thc skinplatc dcforms 2.6 mm to、 vard the downstrcaln side,and the top centcr dcforms toward

Figure 6:Static defor7nation by FEM analysis.

thc upstream sidc. The portion which has no dcfonnation (diSplacement is zero)iS ncar ule 2nd radial arm.This stdic

dcforlnation distribution is h good agrecmcnt with similar FEM allalysis rcsults for thc Folsom Dam Tahter gatc[see

Anami(2002)]. 、 vcrc Subsequently, the natural vlbration charactenstics air skinplate strealmwise bendヒ 唱 vibration.Thc resultant bcnding vbration mOdc of he skinplatc is shOwn in Figure 7,whcre he bcl■ dmg

dctcrmhed via FEM analysis for thc m¨

vibration modes in dle spanwisc direction arc shown on the

Hght side. These mode shapes compare very wen、vith ulose for the Folsom I)alm Tainter―

→>圧

gatc,sho、 vn in Figurc 5.

Thc natural frcquency of tllc skinplatc bcnding vlbration

→圧



//´

Figure 7:Vib「 ation mode shape of3-D equivalent rnodel

by FEM,for skinplate streanlwise bending vibralon.

for thc 3-dilncnsional cquivalcllt modcl gatc as dctcrnlincd by

thc FEM simulation was 56.O Hz This valuc is about 2.l times highcr than thc incasured natral vibration tcquency of 26.8Hz for tllc Folsom Dam Taintcl― gatc. Howcver, in tcsts with

rumulg watcr,dlc model frcquency decreased to almost ule samc valuc as for the Folsom Daln Tdnter― gate due to■ e addcd mass ofthc water.

cOuld not be rcalizcd at an fOr thesc highcr frcqucncy vibrations. h thc model cxpcrilllcnt,it is ncccssary to sct in― cqucncy water natural vibration tcqucncy Close tO tllc 6.O Hz■

Of thc prototypc

Of coursc, it is nccessary to corrcctly

rcproducc thc strcalnwise bending vibration of the skinplate Thus,to this end,a3-dimensional rnodel gate、 vith less rigidity,

Based on this FEM simulation, thc 1/13-scaled 3-D cquivalent mOdel gate、 vas constructed.A little creativlty was required for f破 hg the O.71Ylm thick skinplatc on thc cttcular―

arc ribs hitiany loo temporary五 vets wcrc uscd to attach ulc skinplate to thc五 bs Thcll tllc skhplatc was soldcrcd to hc 五bs to providc thc dcsircd rigidity Thc dOwnstrcam alld sidc vicws of llc 3-I)cquivalent modcl gate,installed in the flumc,

Copシ Tight《)2006

byASME

(a)UpWard view

(b)Side view

Figure 8:Photo ofthe 3¨ dimensional equivalent model gate insta‖ edin the lume. are shown in Figurc 8 1n order to prevent distortlon ofthc gatc

structurc,crosshg wircs arc s“ tchcd bctwccn thc radial arlns. ͡ヽ c3‐ D equ市 alcnt modcl gatc has a total mass of 42.45 kg,as own in Tablc l,m which thc masses of maJor parts arc also

Tab!e

l:

Component

masses

SkinDlate

presented.

WATER FREE V:BRAT:ON TEST Fttst,thc in― air

Radial arm f× 8)

T血

natural strcalnwisc vbradon tequcncy of

the skinplate, ΩnaΨ , was mcasllrcd by in_dr modal analysis

(impulS市 C hammcrmg),yieldil■ g a frequency of56 0 Hz,which agrccd with ulc FEM prediction

Thc 3-D cquivalcnt modcl gatc was hstalled in thc nume to conduct vlbradon tests,as shown in Figure 8(b).The test condtions and mttOr mechanical constaltt are sunlmarized m Table 2. The m_water natural streamwise vbration frequency Ωnwv was lowcred to 6.28 Hz witll the gatc submergence dcpth of l.12 m, which was vcry close to the in― water natural

vlbration frcqucncy of 6.46 Hz identiflcd for Folsom Dam Tainter― gatc by our theorctical analysis.Thc■ sl士plate is inclined at 2 1°

oorjust undcr thc

so that with■ c static deforlnadon of

thc skinplatc with hydraulic pressure,the gate is the neutral positiOn between acting as a press¨ shut devicc and acting as a press― open device,that is,伍 e so‐ called geolnetric press¨ shut ͡ glc was sct as O°

In order to a● ust cOrrectly the wholc gate rotational vibration icqucncy ΩnwO, lcaf springs wcre connected to thc

1/13¨ scale

3D

7.04 7

Circular… arc rib r× :N¨

of

equiva!ent model of Folsom Dam Ta Ⅳ[ass「 k2

On hub(× 2)

17.29 16.88 1.24

42.45

Total

丁able 2:Test conditions and mechanical constants for 1/13-scale 3D equivalent model of Folsom Dam Tainter¨ 3-D model oatc

l H5m

dcpth ofupstrcarn channcl

B dcptll ovcr thc gatc bolttom

inclination anglc ofgatc bottom to water surfacc

natral vib icq Of upward vib in―

air natural vib frcq ofstreamwisc vlb

in―

atr damoing ratio for upward vib

0005m 1110m

Os

vancd

Qwn

Q

614 Hz

0012

ill alr damoing ralo for strcamwisc vlb rotatlon ccntcr hclght

Rs

orcss― oDen

0

angic

rotation rasius ofskillplatc lower end



0626m L“

m

、 vire cables used for raishg tlle gate, as shown in Figure 9, 、 vhere t、 vo guide bars are also attached along with ■c wirc cables tO prevent any streanlwisc swinging motion of tllc lcaf spring

Two no‐ contact typc dittlaccmcnt scnsors(PU-40お m AEC Co)wCrc positioned on thc back¨ sidc strcamwisc― cellter bottom of thc skillplate to llllcasure the wholc gate rotational

vLration arollnd the trunnion ph and the skinplate streamwise bcnding vibration

From the analysis of Allami&Ishii(2000,2003)and Allalni(2002),violent coupled¨ mode sel■ excited vibration are cxpected to occur 、 vhen the m― water natural vibration frcqucncy of tllc skinplatc strcamwise vlbration, ΩnwΨ , is slightly lcss than thc natural vibra■

on frequency of the whole

gatc rotational vibration around the trunnion pul,Ω nw。 .Based

Figure 9:Leaf spring and guide bar.

on these analyscs,the、 vholc gatc rotational vibration frequency

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[mm]

卜J

Strcalnwise Vlb

︲  2 一

Ωw√ 628Hz ξ w√ 00152 84[S]

[mm] [mm]

[nlm] Up‐ and― downward

Vib

ΩwΨ =628Hz ξ wΨ =0012 84[S]

[mm]

ligure 10:Vibralon test results wlh 1/13… scale 3-D equivalent model gate for vibralon waveforms and tralectoHes at skinplate bottom center

ΩnwO was attuSted tO be 6.62 Hz,slightly larger than the skinplde streamwisc bendhg vibration ttequency ofΩ nwΨ =628

丁able 3: Non… dimensional factors of lノ 13-scale 3D equivalent rnodeli needed for theoretical calculations. 3-D model

Hz,yielding a frcqucncy ratio of skⅢ latc Streamwisc natural vbration to wholc gatc natural vibration,γ O.99。

Ⅳ(≡ ΩttΨ /Ω aO)Of

With this condition,a very mtcnsc dynamic hstability

、 vas observcd, as shown in Figure 10, where■ e upper dlnc history is the skinplate radial vibration and the lowcr dme history is the skinplatc tangential宙 bra■ on,represcnting the skinplate streamwise vわ ration and the whole gatc vbratlon around■ e trunnion pin,respectivcly. Thc violent sel'cxcitcd

vibrations were spontallcously hduccd. At an amplitudc of about 13 mm thc 3-D cqu市 alent modcl gatc stluck J■ c chalmcl ■ooち

and was prcvcntcd iom any hrthcr incrcasc in

dcptt ratio(β

≡d/d7)

reduced rotatlon ccntcr hcight(rs=Rs/dO) rota6on radius ralo(、

momcnt― O■ inc■ ia

≡Rs′1に )

ratio(α

I三

rsa

I./1n)

moment‐ o■ inc■ ia ralo ofskinplate(α lvttllJO/1∂ waterto gate mass ralo(α

‖≡ρd02wO/(Im/Rs2))

072

instantaneous■ ow― rate― coe価 cient DreSSure correction coeficient basic Froude numbcr tRI=2■

Ω,v×

δ。

1

Vd。 /g

almplimde.The flrst point of great signiflcancc h this papcr is

that it has been clcarly demonstratcd with thc 3-D equivalent model gate that Taht∝ gates are rcadlv susceDtible to intense ・ 101cd― mode sel■ cxcited vbration. Both vibratiOn icquencies Ωwv alld ΩwO m Figurc 10 wcre

cxactly 6.28 Hz,synchronized wi■

thc skillplatc strcamwise

namal vibration.Thc cxcitation ratios(negative damping rdios)ξ w and

wO mCasured from small amplitudc pOrtions of ξ

thc vibration tilnc historics werc O.015 alld 0 012,rcspcctivcly

Thc incasurcmcnt location of vbration was at tllc bottom spttwise center of thc skmplate.Thc vlbration tra」

ectories are

shown h■ e

%=論

+Cw



where γttw(≡ く ヽΨOaΨ )represenヽ

tlle i‐ wtter

to m― air

vbradon icquency ratlo of the skinplate sttcalnwise vlbration. Thc nuid excitation ratio

ξfvjtakes a valuc ofO.132.

Similar measurements were made for various vibration tcqucncics Ωnwo Of wholc gate rotation around thc trunnion pin Thc tcst conditions alld mcchanical constants are shown in Tablc 2

right sidc ofthe Fturc 10・ The upper廿 句ectOry shows comparat市 cly small vibration,while dle lower one

THEORETICAL PRED:CT10NS

shows comparatively large vibration.The angle ofthe slope of thc traiectOry iS exactly 45° against the mclined nOorjust under

vibration of non― ecccntric Tainter gatcs was developed by

tllc skmplate,thus effect市 ely supplゾ ng energy from the nuid

Anami&Ishii(2000),WhCre it was assumed that the skillplatc

mOtion to the gate vibration, and tllus inducing the obscⅣ cd

rotational streamu7isc bcndhg vibration can bc representcd by an cquivalent rigid body rotational vibration、vith very small

very intense couplc― mode self‐ excitcd vibration

Thc fluid excitauon ratio ξtt representing the lcvel of dynalnic instability due to fluid motion can be calculatcd iom the following expression:

Thc heoredcal analysis for the cOuplcd―

mOde scliexcited

amplitude The whole gate rotation around tlle trunnion pul in¨

duces a flow rate variation bencath thc gate, wi■

a

corrcsponding“ now_rat variation prcssurc"which forccs thc skinplatc to vibratc ill thc strcamwisc drcction.subscqucntly,

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2006 byASME

ξ Jへ eJ o ヽ﹄co 稲 一oxo ︻ ︻ ︼ O︼ 〓﹄

Exp3こ ]la:D modcl

● :3D modcl

0.1

こ ャ、

…・:窯 :BI::]

0.05

0。

5

2.5

1.5

γ nw(≡ qwyawO)

Vibrttion icquency rttio

Figure ll:Vibration test result with 1/13-sca:e3D equivalent rnodel gate plotted on the dynamic stab‖ Jiagram,and its compa面 son with the theoretical prediction. the skhplate strcamwise vibration induccs a(`push― and― 壷aw pressure,''which signiflcantly lowers thc vbration icquency of tlle skinplate. Since the gatc discharge is not submerged, no

vortex

phenomena

are

produced,

and

hcncc

thcsc

hydrodynamic prcssures call bc thcOrcticaⅡ y calcula“ d by a potential thcOry dcvcloped by Raylcigh(1883)for diSSipat市 e



With

theorcticaHy

′ 甲

)与



′ 叩

10

whcre thc last terln of tte

五ght― halld sidc reprcscnts hc signiflcant cncrgy sourcc tO cxcitc the strcatnwisc vibration,

duc to flow― rate variation under from the gatc.

wave‐ radiation prOblcms(scC IShii 1992,Anami&Ishii 1998a,

1998b,1999 and 2000b).



[卜

ity criterion

A rnathcmatical approxilnatc analysis, given in Anami& Ish五

derived

cxpressions

fOr

the

hydrodynamic prcssures, the couplcι mode sclfexcited vlbration of Taintcr gates can be theOreica■ y analyzed. Thc coupling of streamwise and whole gate vibrations tllrough thc

hydrodynamic and hertial forccs wtt addressed by Analni& ISh五 (1999)

Thc wholc gatc rotation around the trunnion pin is

independcllt of thc hydrodyllamic 10ad,and is cxcitd by thc

(2000),Can afford the solutions ofthe equations of motion (2)and(3).The nOn_dhensional factors needed fOr theOretical calculations arc listed in Tablc 3. The calculated results arc

shown in Figure ll and represent hc dynanlic stability crilcrion The ordinatc represents the nuid excitation ratio, which is, in other words,the dalnping ratio needed to prevcnt thc dynamic instabillン Ъthat is,so― called thc“ stability damphg ratio''.Thc abscissa is he frequency ratio γll.(≡ ΩnwΨ /Ω aO)of h― water sLeamwise natural vibratbn of he skinplate to thc in―

air

inertial torquc Of thc skmplat itsell thus bcing rcduced to thc

whole gatc rotation around thc trunnion pin. Thc soLd linc

foHOwing non¨ dimensional cxprcssiOn:

rcprcsents the stability damping ratio,calculated for thc prescnt 3¨ D equivalcllt model gatc■ Om thc heorctical analysis

0″

+2ζ α 0′ 。



+0=―

(2)

Ct* `LΨ

where O and Ψ rcprcscntthe rcduced counter― clockwise rotation

angle of the wholc gate around the trunnion pin and of the skinplatc around its rotation ccnter,respcctively(sec Figure l)_ In Equauon(2),■ c daSh rcprcscnts me der市 at市 e with respcct

to me reduced tinle.This study assumes sma11-amplitude vlbrations,thus neglecting the hydrodynamic pressure torque due tO tlle smJl shi■ of thc skmplatc ccntcr iom thc ttunnion one. In contrast, tlle skinplate rotatiOn k1/ in thc streamwisc d士 ection

is excitcd by both he hydrodynamic load and tl■ e

secondary torque of inertia,thus being suttcct tO thc followmg non¨ duncnsional expression:

+δ ′ α +√ ιδ △ α 勧∫″ 〔 Ψ 」 ′ Ψ “ /た

small flncd circlcs h Figure ll The sccond pOint of great signiicancc in this papcr is that tlle 3¨

)‐



D cqu市 alent modcl gate

tcst rcsults ovcr this region of intcnse dvnalnic instabllitv show

plose agreement with the theoreticallv predicted stabilitv criterioni tllus irther validating our■

eoretical predictiOn.

In addition,Figure ll shows 2-dimensional model gate test results as unf11led data points and tlle cOrresponding theOrctical

results by dotted lincs. The thcoretical analysis has been validated not only fOr the 3‐ diinensional lnodel tcsts,but also

by compaison wi■ results from a 1/31-scaled 2‐ dimcnsional

modcl gatc[SCe Anami&Ishii(2003)]and With ill― Tainter‐



脅呻 卜辮 力 需

The mcasured fluid excitation ratio data calculatcd ushg

Equ面 on(1)and the mcasured vibratiOn data are also plotted as

scalc

gatcs under practicai ncld usc[SCC Anami,Ishii&

Кhisely(2004a),(2004b)].

Copyright◎ 2006 by ASME

丁able 4: Maior Spec雨 cations of the fa‖ ed Tainter

MttOr SpCCiications of tllc failcd Taintcr gatc at thc

Folsom Dam are shown in TJ91e 4,m which mcchallical

w剛 o ︲ F

DYNAM:C INSTAB:L:TY OF THE FOLSOM DAM TAINTERⅡ GATE

B 0.

1402111

10

0762m



13 26 nl

Rs

61×

10° kσ

m

96m

Ⅵcchanica

(ち 。

6 88 Hz

Ω

26 9 Hz

constants

0012 0002

Ω in¨ air constants such as tllc m_air vbration frequcncics ∼ dalnping ratios ζa alld rot江 lon center height Rs have been dcterinined by flcld vibratiOn tcsts on a rcmainhg gate of thc

Table 5: Nondimensional pa「 ameters for theoretical

salnc design as he failcd gatc[see IShii(1995a,1995b)for

calculations

dctails].The gatc was installed over a cun/cd dam∝ inclination angle and a gcomctrical prcss― opcn angle of 85° .(ThC pК SS― open anglc is thc allgle betwccn thc crcst tangent alld tlle norlnd to tlle skinphe secant)The aVerage gatc opcning B、 vas 0 762■ 1,or 5 70/。 of the gatc submergencc dO of 13.26 m Thcrcforc, tl■ c assumption of a small gatc opening is justiicd.Moments of incrtia of the wholc gatc and

O、

亀 `

cst with a

9.5°

ジwhOlc gate,α I(≡ Iv/1o),whiCh affects tllc coupling levcl of the streamwisc vlbration of tlle skinplatc with tlle rotationd vibration about thc trunnion ph, is comparativcly slllall at 0 123 Thc water― to¨ gttc mass ratio αΨ (≡ ρdo2ⅥzO/(IΨ /Rs2)) rcpresentatlvc of a light、 vcight design,takcs on a largc valuc of 129,、 vhich rcsults in a signincantly largc in― watcr rcduction of plate in¨ thc sttcamwisc vlbration frcqucncy rclativc to tllc skil■

2 ヽ〓 o ︼む 8 3 o C C 罵 と

skinplatc,I,were calculatcd numcrically. Non― dilncnsional paramctcrs ncccsstty for tl■ corctical calculations ofthc dynanlic hstability arc presentcd h Tablc 5. Thc reduccd hcigllt oftllc rOね uon ccntcr,rs(≡ Rs/d。 )takcs on a 外くluc of 0 72 Tlle momcnt― o'ulcrtia ratio of thc skillplate to

Basic Froudc numbcl F。 Figure 1 2: !n― water to in― air vibration frequency ratio

ofthe skinplate streanlwise vibration of Tainter gates

takCS On a lagc valuc of 196 Thc

instantancous flow― ratc coc伍 cicnt cf was lncasurcd by modcl

cxpcrimcllts to bc 0 72[scc Anami,Ishii&Takano(2003)for dctails] Tl■c

dynanlic instability of the failcd Tainter gatc at

strcamwisc vbrdion,Ω





Folsom Daln can bc tl■ eoretically calculated[sce Anami(2002) and Al■ alni&Ishii(2003)for details].As part of such an analysis,thc in― water to in― air icqucncy ratio of the skillplate

/Ω 型 (=F″ 0)muStbC determmcd and

thc rcsult is sho、 vn m Figurc 12.

In this flgurc,the abscissa is

tlle basic Froudc number FO and calculated results[re shown by

Ч              Ю

島 =√ 0/gOッ

き︶2︸ 8︶^ ︱ ∞口五日” ︸ 一む嬌増︺ ∽

air valuc. Thc basic Froudc number FO dcflncd by

● :Folsom gate datunn on its incipicnt failurc

Stablc

Unstablc

solid and dottcd lhes.Small ordinate values in his flgurc

を` ぅrescnt signiflcant addcd mass effects for tl■ e in― watcr gate _υ ration. Expe五 mcntal tcst rcsults arc also plotted,showing good agrcement bet、 veen tllc correspondillg thcoretical rcsulヽ and test rcsults for modcl gatc tcsts[scc Anami,Ishii&Takano (2004), for ill¨ Scaled Tainter¨ gatc “A"[Anami, Ishii & Knisely(2004b)],and fOr full― scalcd Taintcr― gatc“ B"[Allami, Ishii&Kniscly(2004⇒ ]FЮ m Figurc 12,tllc icqucncy ratio ΩwΨ /Ω av Ofhe Folsom E)an gate is O.24 at FO=196 This rcsult suggests F=47 2 fortheね iled Tamter gatc attlle Folsom dam Subsequcntly. tl■ e dynalnic stability critcriOn curvc for

solid line in Figurc 13, wh∝ c thc ordinatc rcprcscllも

tllC

required dampillg ratios ζcΨ and ζco,llCCCSSary for colmplete dynanlic stability agamst thc fluid cxcitation.

Tllc abscissa is

Tainter gate atthe Folsom Dam. skinplat stlcall■ wisc vbration with a tcqucncy of f2nwΨ

胤譜胤 憮 Iキ慎露

]庶 auttstti躍為躍

structural damping.For the failed Folsom Dalln gate,he、 vhole gatc vibration tcqucncy about thc trunnion ph is(2aO=6.88 Hz (SCC Tablc 4),witll a cOrrcspondlllg valuc of

γnw=0 94 Forthis

flcqucncy ratio tllc sttcalnwisc vbration dal¬ plllg ratio

ζ =0002(scc

γnw=1.0 rcprcscnts hc rcsonance statc of thc couplcd vibraion lf γnw is

stability.

smallcr ulan l o,the coupled vibratlon is synchronizcd、



=646

cΨ ShOWS a largc pcak with values beyond thc expected ζ

the tcquency ratio γmv, WhiCh is dcflned by thc ratio of thc skinplatc m_water inhcrcl■ t vlbration tequency ΩnwΨ to tllc wh01c gatc in¨ air rOtational vibration frcquency,Ω aO.

γ nw(三 Ω nWyΩ aO)

Figure 1 3:Dynamb instab‖ ity at F=47.2 1brthe failed

tめ

c

F=47.2 ofthc failed gatc can bc calculatcd and is givcn by tl■

Vibration icqucncy ratio

Table 2-4)is p10ttCd by tllc black dot,which is aΨ far smaller than tlle requttcd damping ratio ζcΨ =0 051 for

vith thc

Copyright◎ 2006 by ASME

Anami, K.& Ishii, N., 2000,Flow― hduced Dynamic

As shown here, it has been prcdicted by our■ eorctical analysis tllat ic Folsom Daln Tainter‐ gatc at its hcゎ icnt failurc 、 vas locked in an intense dyllamic hstability which appears■ dle宙 bralon frequency ratio vnw sliglltly smaller than l.0,tllus hducing the serious failure[refer tO Anami,Ishii,

Knisely&Oku(2005)]

lnstability C10scly Related to Folsom Dam Tainte卜 Gate Failure

Analni,K&Ish五 ,N.,2001,Analyses of FIV of Folsom

ふ 月 漁′ 鵬 9」 磁 ″馴課協鴨嚇期声L協 〕 競芳/ θД ″

CONCLuS:ONS

&■

in Califomia,ル r FJa″ ルあ εθ ′ ηら″″ ο″ 脅グsi S.Zブ αあ ブ

S″ ンら′ り,205-212,Balkema.

″あ″ο″グ J77`OEC仏 V01.2,pp.165-181. ami,K.&Ishii,N.,2003,Model Tests for Non―



Al■

h this study wc have dOculnented the existence of a mechallism for thc coupled‐ mOdc selicxcitcd instability of T」 nter― gatcs that has nothing to do with the eccentricity of the

mounting, using a 1/13-scaled 3-D model gate of the failed

Folsom Darn Tahter gate. h addition,wc havc provcn th江 the failurc of Folsom Daln Tintcr_gate ccrt壼 nly could havc been causcd by ulis coupled‐ modc self_excitcd instability mcchanism This nc、 v dン咀amic hstability necds to bc

Ecccntricity Dynarnic lnstability Closely Rclated to Folsom Dam Tainte■ Gatc Failure, シ ″′οSゴ ク″ 0″ FJa″ ― Jη ′ ′ ′ g,P′ο “ Jο κ ″ gsグ ИSに E Pressン r`フlθssθ おα んb″ ′ 2θ θ ε θ ″ノ ″ グ

,V01465,213-221. ″ 花 のψ ′ `46θ Anami,K.,Ish五 ,N.&Knisely,C W.,2004a,Validation

of Thcoretical Analysis of Tamtcr Catc lnstability l」 sing Full― Scalc Field Tcsts,Prac.グ 滋ι∂滋 動 たr″ α″οκα′Gο ψ ra“ ο′

″ンググε ″ んbκ ′ ′ ′γ c″ んら ″″ο ″,ル r Лο ブ ο E&И χなα,Fり ,V01.1,pp.409-414

F/ο wIれ

g′ の Zα ′

Allami,K, Ishii,N.& Knisel光

recognizcd alld precautions taken to prevent gate failllre by this

SI Dθ

`〃

C W.,2004b,Field

mechanlsm.

Vibration Tests and Dynamic Stability Analysis of A Full―

Thc coupled― mode self― cxcitcd instability is very much like a canccr or othcr discase that inay lay dormant for ycars Lcforc manifesting itsclf with potcntiany dcadly rcsults.This

α″″P″ 滋g Cο

bccausc the mccllanisnl usuaHy requircs a triggerhg dlsplaccmcnt hat is sufflciently large to overcomc thc

mcchanical damping[scC Anami,Ishii,Kniscly&Oku(2005) for dctai].The sOurcc of this triggcring displaccment may bc as apparentlybenign as corrosion on the trunnion pin or may bc

as overtly dangerous tt an eartllquake.

resultant

vbrations

can

be

avoided

h any event, tllc

through

propcr

design/retrofltting of the skin plate to give it sufflcient rigidity

、 vater bending mOde frcqucncy of the skin platc call ncvcr become coincident、 vith the rotational icquency of so that the in¨

tlle gatc as a rigid‐

body

Wc would ask dlat ule rcwccted hydraulic sttucturcs

authoHtics h tllc world carcfully rcvicw the matc五 prcscntcd,

makc

Scalcd Tainter― gate,P/ο θ_0′ И鋸

warrantcd,and aftcr tllat rendcr a judgment that will let the

hydraulic structurcs community bccome incrcasingly aware of this dynamic hstability that must be considered for every Taintcr gate.

/餌

惣ysθ Js 劇ИE Prass′ r` フ

,V01.488,pp.81-88.

"cι Anami, Ishii, Knisely & Oku, 2005, Hydrodyllamic `嘘

Pressure Load on Folsom Dam Tainte卜 Gatc at Onsct ofFailure duc to Flow― Induccd Vibrations,Syl■ )osium on Flow― Indu∝ d Vibration-2005,PЮ c.グ ИttE Prの sン ″ フ sθ rs α″グP″ Йg `ω ′ の政′ ε .

`刀 Anami, K, Ishil, N & Takano, W., 2003, Elmptical

cvaluation mcthod of hydrodynanlic prcssures hduccd by vibratlons of inclined circular― arc skmplatc of Taintcr gates, ′グ S″ α′″′E4gJi4gettg,Vol.49A,pp.645-651,(in "α “ Japanesc).

Jο 夕

Anami,K,Ish五 ,N,&Takano,W,2004,Thcorctical

Allalysis of Flow― Induced Streamwise Rotating Vlbration of

hclined Circular_Arc Rigid Skinplate and lts Veriflcalon by ,ルα″″ αJο sげ 滋`J駆 ,Ser.c,70-690, pp 385-393,(h Japancse). “

Model Tesヽ

Anami, K., Ishii, N

als

any additional investigations they feel



and Yalnasakl, M, 2000,

Hyttodynamic Pressure lnduced by Rotaing Skinplate of 「 ο3ヽL,Scr B, Folsom Dam Taintcr― Gate,r“ ″sα αノ Oκ Sり `力 66-652,pp.3116-3123,(in Japancsc) IshH, N., 1992, Flow― Induced Vbration of Long¨ Span

Gtts(Part I:Modd Dcvclopment),力

′綱α′グ E7タ ブ 赤 α″′

θ ユ 125%ぶ γ 亀 絲絆 留%?留 ::rα 解 た 趙 ヽ RcclamatiOn,(Aug.24,1995). ´ Lぜ 決Ъ :協 譜 )i鵬 :段題 LEI絆 糧 ピ ッ認 棚配 動s別)悦 λ 7鶴 慮鴨 3冨∫ 夕鶴溜 鰍 : to U S Bureau ofReclalnation,(Nov.8,1995). S″ frε rtrres,V01.6,No.5,pp 539-562.

ACKNOWLEDGMENTS

s“



´ .ants¨ in― Aid for Scientiflc Research from Japan Society for the Promotion of Science.The autllors' extend thcir thanks to

Munchiro Shirasaka, YoshitO Uclnura, Satoshi Sasaki and Takashi Murata,who assisted wi■

ic model expcriments.

F/″ ブ ″Eれ g″ θι万″g,Ser.I,Vol.99,No.4,pp.699-709

Analni, 2002, Flow― Induced Coupled― Mode Self Excited

Vibration of Largc― Scaled Tahter‐ Gatcs(h Japancsc),

CO″ ″ン4ブ ″ο″ Dお ser如 万ο″ s′ b″ げ ι たd ゎ Osα tt E′ θε″ο― “ νarsブ ヶ,March 2002 Uれ ′

& Ishil, N, 1998a, h¨

Wat(汀 Stream、visc

Vlbra● on of Folsom Daln Radial Gates in California,P74ο ε グ

′99∂ ИS2四 P/P Cο グ ,V01.363,87-93(Sall Dicgo,USA) Al■ alni,K.&Ishii,N.,1998b,In― air and h―

Vibra■ ons

Water Natllral

of Folsom Daln Mial Gates in Califomia,動 」 ″ α′Mβ cみ α4ブ ′脅′ 「И〃なο4工″ り,29-34,Balkcma `η “ Allallli,K.&Ish五 Induced Coupled― Mode ,N.,1999,Flow¨ Vibration of Folsom Dam Radial Gates in Califomia,P′ οε グ ′999 ИSi″ P/P Cο び ,V01369,343-350(BostOn,uSA).

Eψ αれ

Imaichi, K, 1977, hstability of Elasticany

乃θИSME,力 ″ α′グ thc Rcservoir of a Dam,磁 ″sα θ′ わ″sグ ′

REFERENCES

Anami, K

Ish五 , N. 2財

Suspcnded TahtcFGate Systcm Causcd by Surfacc Wavcs on 1sh五 ,N



& Imaichi, K, 1980,Dyllamic lnstability of

Tamter_Gatcs,In:P′ α たαノEη,θ ″ノ θ″θ

/ib″ ′ ゴ οsβ ″Sr Ⅳz′ あ`′s訪 銑 “ Bcrlin,Springcr Vcrlag.

グツθιグ Dり ,pp452460,

142Jtt FJa14 J″

E&Rο εわ “ッ

`″

Naudaschcr,E, 1984,Field Tests on Natural Vibration Modcs ofA Tahtcr Catcs,In:Cん α″″θJs α4グ Cみ ακκι′ Ish五 ,N. `と

″ ″″S″ ンθ ω Cο ″ ″″

Edr K/乃 rS“ ノ ′ 〔 り,pp.209-222,Springer―

Vcrlag.

khii,N.and Naudaschcr,E.,1992,A Design Criterion for ″α′ F械通sα ″ ′ Dynamic Stability of Tainter Gates,カ ン θ s,V016,Nol,pp sr″ ε ′

67-84. `′ `ッ Rayleigh,J W.S.,HThc Theory of Sound",2,(1945), pp4-8.New York:Dover.

Copyright《 )2006

by ASME