field test validation of analytical model for …

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Int. J. Mech. Eng. & Rob. Res. 2014 Keiko Anami et al., 2014

FIELD TEST VALIDATION OF ANALYTICALMODEL FOR VIBRATION CHARACTERISTICS OF A

FLAP GATE UNDERGOING SELF-EXCITEDVIBRATION

Keiko Anami1*, Noriaki Ishii2, Charles W Knisely3 and Yukio Matsumoto4

*Corresponding Author: Noriaki Ishii, [email protected]

Experimental modal analysis, using an impact hammer and accelerometers, was conductedon a full-scale flap gate with a height of 0.963 m and a span of 14.8 m to determine its in-airnatural frequencies, mode shapes, and modal damping. Subsequently, the in-water self-excitedvibration characteristics of the gate (without any spoilers) were recorded using the sameaccelerometers. The major in-air vibration characteristics (the mode shape, frequency anddamping ratio for the damped vibrations), as well as the major in-water self-excited vibrationcharacteristics (the excitation ratio and frequency of the self-excited vibrations in-water) aretabulated. In parallel with these experiments, calculations of the inherent in-water vibrationfrequency of the gate using a potential flow theory, based on input from the in-air modal testing,are presented. Comparison of the calculated inherent in-water vibration frequency with themeasured frequency of the in-water self-excited vibration confirms the validity of the presenttheoretical analysis.

Keywords: Flap gate, Flow-induced vibration, Added mass, Modal analysis, Self-excitedvibration, In-water frequency

INTRODUCTIONAmong the many hydraulic structures withwhich engineers deal, the hydraulic gate is oneof great significance due to the volume of water

ISSN 2278 – 0149 www.ijmerr.comVol. 3, No. 4, October 2014

© 2014 IJMERR. All Rights Reserved

Int. J. Mech. Eng. & Rob. Res. 2014

1 Department of Mechanical Engineering, Ashikaga Institute of Technology, 268-1 Omae, Ashikaga, Tochigi 326-8558, Japan.2 Department of Mechanical Engineering, Osaka Electro-Communication University, Osaka, Japan.3 Department of Mechanical Engineering, Bucknell University, Lewisburg, PA 17837, USA.4 MeiwaConsulting Company, 474 Minami-mati, Nabari-city, Mie Prefecture 518-0729.

it retains for flood protection, municipal watersupply or irrigation purposes. The gate can beexposed to enormous hydraulic forces and isthus potentially subject to vibration. The loading

Research Paper

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on the gate due to the inertia effectsassociated with the streamwise vibration of thegate can quickly lead to gate failure. Thestreamwise gate motion must also acceleratea tremendous mass of water in contact withthe gate—the so-called added mass. Thisadded mass also lowers the natural frequencyof the gate and can lead to coalescence ofinitially disparate streamwise and vertical gatevibration frequencies. A case in point is thefailure of the Folsom Dam Tainter gate in July1995. Extensive investigations of this gatefailure have been undertaken by Ishii (1995a,1995b and 1997) and Anami and Ishii (1998a,1998b, 1999, 2000 and 2003). Their resultsindicate that the hydrodynamic loading due tostreamwise gate motion coupled with thevertical vibration of the gate most likelycontributed to the failure.

It would be very useful to develop an analysismethod, which would permit the calculation ofthe large magnitude hydrodynamic pressure

acting on gate when it undergoes self-excitedvibration. Ishii (1992), Ishii and Imaichi (1977),and Anami et al. (2012a and 2012b)developed an analysis method, which permitsthe calculation of these hydrodynamicpressures using potential theory for thedissipative wave radiation problem.Laboratory-scale model experiments (seeIshii, 1990, for example) have confirmed theanalytical results. However, there has been noconfirmation of these analytical predictions ona full-scale gate with overflow. An opportunityarose to undertake an experimental study ofthe vibration characteristics of a full-scale flapgate. The results were used to validatepredictions made using the analytical method.

The field test was carried out on a flap gate,shown in Figure 1, with a height of 0.963 m, aspan of 14.8 m and a mass of 3474 kg usedto dam a river for irrigation purposes. The damconsisted of two such gates, located side-by-side with a center pier. By lowering the one

Figure 1: Cross-Sectional View of the Flap Gate in the River Installation

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gate completely, the upstream reservoir wasemptied and the other gate could beinstrumented for in-air testing. The first seriesof tests were in-air experimental modalanalyses, employing an impact hammer andaccelerometers. The spanwise center of thegate was struck with a 5.4 kg impact hammer(PCB Model GK-086B50) and the vibrationalacceleration response was measured atmultiple locations on the gate surface usingLion LS-20C accelerometers.Theaccelerometer data was then processed usinga digital computer to yield the modalfrequencies, modal damping ratios and themode shapes of the in-air natural vibration.

The second series of tests, thoseconcerned with the in-water self-excitedvibration of the gate, were conducted with theupstream reservoir restored and with waterflowing over the instrumented test gate. Forthese tests, the spoilers that had been addedto the gate crest to prevent vibration wereremoved. The flap gate quickly becameunstable and experienced intense self-excitedvibration. The accelerometers produced real-time records of the growth of these of self-excited vibrations. These real-time waveformscould then be processed to yield the frequencyof vibration and the excitation ratio (thenegative of the damping ratio).

Thirdly, since the lowest frequency naturalvibration mode in air corresponds to the self-excited in-water mode, an analysis of thehydrodynamic pressure was carried out for thismode. The frequency of self-excitation in waterwas calculated from the equation of motion forthe gate. Finally, the ratio of the frequency ofself-excited vibration in water to the frequencyof the in-air natural vibration was calculated

both theoretically and from the experimentaldata. The comparison was made to confirmthe validity of the present theoretical analysis.

MATERIALS AND METHODSDetailed Flap Gate Description andInstrumentationA sectional view of the full-scale flap gate isshown in Figure 1. The skinplate is rigidlyattached to the torque tube. The skinplatethickness is 9 mm, and the radius of the torquetube is 156 mm. The distance from the torquetube center to top of the skinplate 0.963 m.The skinplate is reinforced byH-beams, 194 mm in height. The whole gatecan be rotated around the torque tube by thehydraulic jack shown as a dashed line in Figure1. The inclination angle of the gate relative tothe vertical direction, denoted by in Figure1, could be varied from 15º to 90º.

In the upstream view of the flap gate, shownin Figure 2, the span of the gate is given as14.8 m. Also evident are the twelve H-beamsreinforcing the skinplate. Figure 2 also showsmany spoilers that were installed along the gatecrest to attenuate self-excited vibration. Thesespoilers were removed during testing of thein-water self-excited vibrations. Unfortunately,they were not removed until after the in-airmodal analysis, requiring a scaling ofmeasured natural frequencies.

The natural vibration characteristics of thegate were determined by in-air experimentalmodal analysis, employing an impact hammerand accelerometers. With the upstreamreservoir emptied, the gate was struck with a5.4 kg impact hammer (PCB GK-086B50)which was instrumented with a forcetransducer (PCB GK-206M06). The location

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of the blow was chosen near the spanwisecenter as indicated by the mark in Figure 2.The impact was in the upstream direction. Theresultant acceleration response wasmeasured by 6 servo-type mechanicalaccelerometers (Lion LS-20C). These 6transducers were sequentially positioned at the36 points indicated by the small black circleson the H-beams in Figure 2. For laterreference, these points are denoted by a letterand a number according to the followingscheme: A, B, C in order descending from thetop and spanwise locations 1, 2, ..., 12 startingfrom the left side in Figure 2. The impacthammer force and the acceleration responseswere recorded using a digital data recorder(TEAC RD-135D).

All data were analyzed on an FFT analyzer(A and D AD3525). Measured impulsive forceand the resulting acceleration response atPoint 6C are shown in Figures 3a and 3b.Theacceleration power spectra were divided bythe impulsive force power spectrum, resultingin the transfer functions at Point 6C, as shownin Figure 3c. This transfer function is theaverage of 3 sets of impact and responsedata.

The calculated transfer functions for all 36measurement points were input into a modalanalysis software package (AD1461 by ZonicCo.). The software package calculated anddisplayed the Inverse Modal IndicatorFunctions (IMIF) shown in Figure 4, which isdefined by the following expression:

n

ii

n

iii

T

TTalReIMIF

1

2

10.1 ...(1)

where Ti represents a transfer function. Thesecond term of Equation (1) corresponds tocos, where is the phase-lag of the vibrationresponse relative to the excitation force andtakes a value of 90° for resonance. Therefore,the peaks of the IMIF show a level of intensityat potential natural frequencies of the gate. Thefrequencies of the IMIF peaks were selectedand the software attempted to calculatecorresponding mode shapes. All peaks do notnecessarily result in reasonable mode shapes.However, a fair number of the IMIF peaksresulted in meaningful mode shapes. Thepeaks corresponding to these modes aremarked in Figure 3c for the transfer functionand Figure 4 for the IMIF.

Figure 2: Downstream View of Flap Gate Showing Impact Location and AccelerometerLocations for In-Air Modal Response and Self-Excited Acceleration Measurement

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Figure 3: Sample of Measured Data Showing (a) The Impulsive Impact Force, (b) TheAcceleration Response at Point 6C, and (c) The Transfer Functions for Point 6C

RESULTS AND DISCUSSIONIn-Air Natural VibrationCharacteristicsAn overview of the frequency, the damping

ratio and mode shape for these naturalvibrations is shown in Table 1. The frequencyof the fundamental mode is 30.5 Hz, and itsmode shape suggests that it is essentially auniform vibration across the entire span of the

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gate. A cross-sectional view of thisfundamental mode is shown in Figure 5. Thisfigure shows both the maximum upstreamand downstream displacements of theskinplate. The numbers near the top of thegate and the letters on the right of the gateindicate the measurement locations given inFigure 2. From this mode chart, it is evidentthat the gate is undergoing rotational vibration.The small black circles show rough estimatesof the centers of rotation on the left side ofFigure 5. The bottom edge of the skinplate

was taken as the zero reference location. Theposition of the center of rotation ranges fromthe center of the torque tube to its outersurface.

Self-Excited VibrationCharacteristics of Flap GateThe bottom of the skinplate was aligned withthe upstream riverbed, as shown in Figure 1.The self-excited vibration tests wereconducted with the upstream water depthmaintained at 0.814 m. An accelerationwaveform of the self-excited vibration with thegate at an inclination angle of 17.2º and anoverflow depth of 16 mm (measured at Point1A) is shown in Figure 6. The vibration grewrapidly and reached steady state after about1 second. The excitation ratio was 0.0072,and the frequency fw was 22 Hz.

Similar measurements were made forgate inclination angles from 15º to about 33º.The measured excitat ion rat ios andfrequencies at two locations are shown inFigure 7. Open circles represent the datafrom Points 1A, while the filled circles arefor Point 12A.

The excitation ratio, shown in Figure 7,gradually decreases, as the gate inclinationangle increases, because the overflow depthincreases and the air cavity volume formedon the downstream side of the gatedecreases. The frequency is locked onto avalue of about 21.9 Hz for inclination anglesless than about 22.2º. The frequency suddenlydecreases to around 19.5 Hz once theinclination angle exceeds about 22.2º. Witha continued increase in the inclination angle,the frequency gradually increases. Theconstant frequency at small inclination angles

Figure 4: Inverse Modal IndicatorFunction for the Flap Gate with the First

Three Modal Frequencies Identified

Table 1: Experimental Modal Frequenciesand Modal Damping Ratios from

Experimental Modal Analysis of the FlapGate

M1 30.5 0.030

M2 39.5 0.019

M3 42.3 0.006

ModeName

Frequencyfa[Hz]

DampingRatio a

Mode Shape

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suggests a feedback mechanism relatedmost likely to the oscillations of the nappe.The mechanism of the nappe oscillation is notthe topic of this paper and is not examinedfurther in this work.

Theoretical Calculations

Hydrodynamic PressureAnalysis of the flow field is needed in orderto calculate the hydrodynamic pressure.However, it is difficult to analyze exactly theflow field for the case in which the flap gate isinclined in the downstream direction.Therefore, the analysis will be confined to thesimplified case in which the skinplate stands

Figure 5: Fundamental Mode of Natural Vibrations in Air Showing Modal Amplitudeand Height of Rotational Center at Each Spanwise Measurement Location

Figure 6: Waveform of StreamwiseIn-Water Acceleration for a GateInclination Angle of 17.2º and an

Overflow Depth of 16 mm as Measuredat Point 1A Showing Growth of Vibration

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vertically and performs the rotational vibrationwith small amplitude, as shown in the left sideof Figure 8. The dimensional coordinate Y isalong the vertical skinplate and is defined aspositive upward from the mean water surface,which corresponds to the dimensionalcoordinate X. The water depth of theupstream reservoir is represented by d0. Thetop of the vertical skinplate above the riverbedis given as dg; the height of the rotation centerabove the riverbed is given as Rs. As shownin Figure 8, the rotation center is lower thanthe upstream riverbed and Rs takes on anegative value.

The pressure change near the water surfaceis comparatively small. Therefore, the effectof the overflow water on the inducedhydrodynamic pressure at larger depths isassumed to be insignificant. Instead, theoverflow water depth was disregarded and itwas replaced by a rigid wall vibrating with thevertical skinplate. The hydrodynamic pressureinduced by the vertical skinplate vibration canbe calculated by applying the potential-flowtheory developed by Rayleigh (1945) andLamb (1932) for dissipative wave-radiationproblems. The validity of the analysis has beenconfirmed in model tests (Ishii, 1990, for

Figure 7: In-Water Self-Excited Vibration Characterized by (a) Excitation Ratio, and(b) Vibration Frequency (Open Circles are for Point 1A; Filled Circles are for Point 12A)

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where0 is the amplitude of the rotation angleof the vertical skinplate. Thus, pb0 representsthe ratio of the hydrodynamic pressure headcorresponding to Pb0 to the vibration amplitudeat the top of the vertical skinplate.

An expression for hydrodynamic pressureinduced by the rotational vibration of thevertical skinplate, Pb, from the previous studyby Anami et al. (2012a), can be reduced tothe non-dimensional pressure pb acting on thevertical skinplate at x = 0. This non-dimensionalpressure pb is given by the followingsuperposition of the acceleration and velocityterms:

FFp

FFpp bpbsb

02

0 ...(4)

example). See the papers by Ishii and Imaichi(1977), Ishii (1990 and 1992), Ishii andNaudasher (1992), Anami and Ishii (1998aand 1998b) and Anami et al. (2012a and2012b) for details of the theory.

The vertical axis y is the non-dimensionalcoordinate along the vertical skinplate:

0dYy ...(2)

Therefore, y = –1 corresponds to theupstream riverbed. The abscissa is the non-dimensional pressure amplitude, defined bythe following expression:

00

00

/

S

bb Rd

gPp ...(3)

Figure 8: Vertical Weir-Plate Model on the Left, and Calculated Valuesof Vibration-Induced Hydrodynamic Pressure on the Right

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where represents the non-dimensionalrotation angle and the prime represents thederivative relative to the non-dimensional time,t, given as follows:

a

Ttand

/10

...(5)

The Froude number F and the basic Froudenumber F0 are defined by

aw gdFand

gdF 0

00 ...(6)

where the in-water and in-air frequencies arerepresented by w and a, respectively.Therefore, the ratio of the two Froude numbers,F/F0, represents the in-water to in-air frequencyratio:

a

w

a

w

ff

FFor

FF

00...(7)

The variables pbs and pbp in Equation (4) arethe non-dimensional pressure amplitudes ofthe so-called “standing and progressivepressure-wave components,” respectively,which are given in the following seriessummations:

211

12),(

12*

4

sj js

bs rr

Fyxp

2*21 /

112

nnjn

1cossin *

*

*

n

nnrs

iiW

y

jj

j

/1

coscos

*

*

...(8)

211

12),( 2

0*

4

ss

bp rr

Fyxp

2*201 /

112

nnn

1cossin *

*

*

n

nnrs

0

*0

*0 1

coscos

Wy

...(9)

in which rs represents the reduced rotationcenter height, defined by

0dRr S

s ...(10)

which takes on a negative value in the presentcase. The value was assumed to be theaverage of the experimental values shown inFigure 5, which was 0.101 m, giving a non-dimensional average height of the rotationcenter of rs = –0.127 which is also indicated inFigure 8.

The variables 0 and j (j = 1, 2, 3, …) inEquations (8) and (9) are the solutions of thefollowing two transcendental equations:

2*00 tanh F and 2*tan Fjj (j = 1,

2, 3, ...) ...(11)

In addition, the functions iiW j / and 0W included in Equations (8) and (9) are

given, respectively, by the following twoexpressions:

*2

**

costan/

j

jjj iiW

and

*0

2

*0*

00 coshtanh

W ...(12)

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where * represents the ratio of the gatesubmergence depth d0 to the upstreamreservoir depth, which has been assumed totake on a value of 1.0 for the present case.

The resultant vector magnitude of

hydrodynamic pressure

22

0 bsbpb ppp of

the standing and progressive pressurecomponents can be calculated. The calculatedamplitudes of the hydrodynamic pressure, pb0,are shown in the right side of Figure 8 in termsof the Froude number, introduced in Equation(6). The Froude number represents thedynamic similarity of flow field and can alsobe written as follows:

gdfF w

02 ...(13)

Inserting the measured frequency of the self-excited vibrations, fw, of about 20.9 Hz, as shownin Figure 7b, with the water depth d0 of 0.814 m,yields a Froude number of 37.9 for the presenttests. The non-dimensional pressure amplitudedistribution for this Froude number is shown bythe solid line in Figure 8. As previouslymentioned, in these calculations the skinplaterotation center height Rs was assumed to takeon the mean experimental value of –0.101 m,with a corresponding value of rs = –0.127(rounded to –0.13 in subsequent calculations).

The hydrodynamic pressure attains itsmaximum value of about 430 (correspondingto 0.43 m of water) near the center of the verticalskinplate when the top of the vertical skinplatevibrates with an amplitude of merely 1 mm.

Added Mass and Frequency RatioThe water added moment of inertia, I, canbe calculated by integrating the hydrodynamic

pressure moment around the rotation centerover the vertical skinplate. As a result, one mayintroduce the non-dimensional added massm, defined by

0

20

20/

WdRdIm S

...(14)

The added moment of inertia was dividedby the square of the rotation radius (d0-RS), toconvert it into the equivalent mass. W0 is thegate span of 14.8 m. The denominator is therepresentative water mass contained in thevolume of d0

2W0. The calculated results for thenon-dimensional added mass as a function ofthe inclination angle are shown in Figure 9a,where the solid line is for the non-dimensionalrotation center mean height of –0.13. Inaddition, the broken lines are for the rs valuesof 0.0 and –0.19, which were the maximumand minimum values of the non-dimensionalrotation center height rs from the experimentaldata. An increase in the inclination angle issimulated by decreasing the height of thevertical skinplate top from the riverbed, dg. Asa result of this slight decrease in dg, the addedmoment of inertia, I, decreases slightly, witha corresponding slight decrease in the non-dimensional added mass.

The in-water frequency of the flap gate, fw,is substantially reduced relative to its naturalfrequency in the air, fa1, due to the added massof the water. The ratio of these two frequenciescan be calculated from the followingexpression (see Anami and Ishii, 1998a):

mff

a

w

11

1...(15)

where is the water to gate mass ratio,defined by

12


20

02

0

/ SRdIWd

...(16)

The numerator in the above expression isthe representative water mass and thedenominator is the equivalent vertical skinplatemass which can be calculated by dividing themoment of inertia of the vertical skinplate, I,by the square of the rotation radius (d0 – Rs)2.The moment of inertia, I, for the present flapgate took on a value of 545.8 kg × m2, resultingin a mass ratio of 15.1. The solid line (for rs =–0.13) and broken lines (for rs = 0 and –0.19)in Figure 9b show the calculated results for thefrequency ratio. As the inclination angleincreases, the frequency ratio shows a slight

increasing tendency, caused by a slightdecrease in the non-dimensional added mass.

Unfortunately, the spoilers shown in Figure2 were still attached when the in-air vibrationtests were made. Subsequently they wereremoved for the in-water self-excited vibrationtests. The measured in-air natural frequencyof 30.5 Hz for the fundamental mode, shownin Table 1, was scaled to 34.1 Hz for the flapgate without spoilers. Dividing the measuredfrequency data shown in Figure 7b by thecorrected in-air natural frequency, themeasured frequency ratio data are obtained,as shown in Figure 9b. It is evident that thetheoretical calculations accurately reflect themeasured data.

Figure 9: (a) Calculated Reduced Added Mass as a Function of Gate Inclination Angle,and (b) Computed (Lines) and Measured (Filled Circles) Frequency Ratio of In-Water

Vibration to In-Air Vibration for the Same Range of Gate Inclinations

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CONCLUSIONThe analytical method for calculating thehydrodynamic pressure exerted on vibratinggates, based on the potential flow theory forthe dissipative wave radiation problems, hasbeen previously validated on various types oflaboratory-scale model gate test. In the presentset of experiments, the analytical method wasindirectly validated for a full-scale gate by usingthe measured frequency data for the self-excited vibrations of the flap gate installed ina river in Japan. Thus, it appears reasonableto expect that using the same analyticalmethod, predictions of the in-water vibrationfrequency and the induced hydrodynamic forceare possible for flap gates as well as for othertypes of gates, such as Tainter gates and long-span gates, in full-scale applications.

ACKNOWLEDGMENTThe authors wish to acknowledge financialsupport provided by a Science and ResearchGrant from the Japanese Ministry of Education.The authors also wish to acknowledge thecooperation and assistance of the village officeof the village of Soni in the Nara Prefecture incarrying out these field studies. The authorsextend their special thanks to Mr. KeijiSakamoto, Dr. Akinori Nakata, Mr. TakuyaTanaka, and Mr. Takumi Tachibana, for theirassistance in conducting the field impact testsand in the subsequent computer calculationsfor the modal analysis.

REFERENCES1. Anami K and Ishii N (1998a), “In-Water

Streamwise Vibration of Folsom DamRadial Gates in California”, Proceedingsof 1998 ASME Pressure Vessels and

Piping Conf., Vol. 363, pp. 87-93, SanDiego, USA.

2. Anami K and Ishii N (1998b), “In-Air andIn-Water Natural Vibrations of FolsomDam Radial Gate in California”,Proceedings of 11 th InternationalConference on Experimental Mechanics,Experimental Mechanics 1, Allison (Ed.),pp. 29-34, Balkema, Rotterdam, Oxford,UK.

3. Anami K and Ishii N (1999), “Flow-Induced Coupled-Mode-Vibration ofFolsom Dam Tainter-Gates in California”,Proc. of 1999 ASME Pressure Vesselsand Piping Conf., Vol. 396, pp. 343-350,Boston, USA.

4. Anami K and Ishii N (2000), “Flow-Induced Dynamic Instability CloselyRelated to Folsom Dam Tainter-GateFailure in California”, in Ziada S andStaubli T (Eds.), Flow-Induced Vibration,pp. 205-212, Balkema, Rotterdam.

5. Anami K and Ishii N (2003), “Model Testsfor Non-Eccentricity Dynamic InstabilityClosely Related to Folsom Dam Tainter-Gate Failure”, Proc. of ASME PressureVessels and Piping Conference, Vol. 465,pp. 213-221, Ohio, USA.

6. Anami K, Ishii N and Knisely C W(2012a), “Pressure Induced by VerticalPlanar and Inclined Curved Weir-PlatesUndergoing Streamwise RotationalVibration”, Journal of Fluids andStructures, Vol. 29, pp. 35-49.

7. Anami K, Ishii N and Knisely C W (2012b),“Added Mass and Wave RadiationDamping for Flow-Induced Rotational

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Vibrations of Skinplates of HydraulicGates”, Journal of Fluids and Structures,Vol. 35, pp. 213-228.

8. Anami K, Ishii N, Knisely C W andMatsumoto Y (2000), “Measured andCalculated Vibration Characteristics ofa Full-Scale Flap Gate Undergoing Self-Excited Vibration”, Vol. 414, No. 1,pp. 23-30, ASME-Publications-PVP.

9. Ishii N (1990), “Flow-Induced Vibration ofLong-Span Gates (Verification of AddedMass and Fluid Damping)”, JSMEInternational Journal, Series II (FluidEngineering, Hear Transfer, Power,Combustion, and ThermophysicalProperties), Vol. 33, pp. 642-648.

10. Ishii N (1992), “Flow-Induced Vibration ofLong-Span Gates (Part I: ModelDevelopment)”, Journal of Fluids andStructures, Vol. 6, pp. 539-562.

11. Ishii N (1995a), “Folsom Dam GateFailure Evaluation and Suggestions”,Report Submitted to US Bureau ofReclamation, August 24.

12. Ishii N (1995b), “Folsom Dam GateFailure Evaluation Based on ModalAnalysis and Suggestion”, ReportSubmitted to US Bureau of Reclamation,November 8.

13. Ishii N (1997), “Dynamic StabilityEvaluation for Folsom Dam Radial Gateand Its Inherent Non-EccentricityInstability”, Report Submitted to USBureau of Reclamation, July 17.

14. Ishii N and Imaichi K (1977), “WaterWaves and Structural Loads Caused bya Periodic Change of Discharge from aSluice Gates”, Bulletin of the JapanSociety of Mechanical Engineers ,Vol. 20, No. 6, pp. 998-1007.

15. Ishii N and Naudascher E (1992), “ADesign Criterion for Dynamic Stability ofTainter Gates”, Journal of Fluids andStructures, Vol. 6, pp. 67-84.

16. Lamb H (1932), “Hydrodynamics”,pp. 398-403, Cambridge University Press.

17. Rayleigh J W S (1945), “The Theory ofSound”, Vol. 2, pp. 4-8, Dover, New York.

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APPENDIX

Nomenclature

d0 water depth of the upstream

dg distance to the skinplate top from the river bed

F Froude number

fa1 in-air vibration frequency

fw self-excited vibration frequency

g gravity acceleration (m/s2)

I moment of inertia of skinplate around its rotation center

Pb hydrodynamic pressure

pb reduced hydrodynamic pressure

Rs skinplate rotation center height

rs non-dimensional rotation center height

Ti transfer function

W0 spanwise length of the skinplate (m)

Y coordinate along the skinplate

y non-dimensional coordinate along the skinplate

water-to-gate mass ratio,

I additional moment of inertia

m reduced added mass

gate inclination angle

excitation ratio

density of water (kg/m3)

phase-lag

0 amplitude of skinplate

field test validation of analytical model for …

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