Emergency gate vibration of the pipe-turbine modeldownloads.hindawi.com/journals/sv/2000/406976.pdf4...

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Emergency gate vibration of the pipe-turbinemodel

Andrej Predina,∗ and Roman Klasincb

aFaculty of Mechanical Engineering, University ofMaribor, Smetanova 17, SI-2000 Maribor, SloveniaE-mail: [email protected] of Hydraulic Structures and WaterResources Management, Technical University Graz,Stremayrgasse 10/II, A-8010 Graz, AustriaE-mail: [email protected]

Received 14 October 1998

Revised 23 November 1999

The vibration behavior of an emergency gate situated on ahorizontal-shaft Kaplan turbine is studied. The analysis andtransfer of the dynamic movements of the gate are quite com-plex. In particular the behavior is examined of the emergencygate for the case when the power unit is disconnected fromthe system or there is a breakdown of the guide vane sys-tem at the moment when the maximal head and capacity areachieved. Experimental-numerical methods both in the timedomain and in the frequency domain are employed. Natu-ral vibrations characterize a first zone, corresponding to rela-tively small gate openings. As the gate opening increases, thevibration behavior of the gate becomes increasingly depen-dent on the swirl pulsations in the draft tube of the turbine.Finally, the data transfer from the model to the prototype byuse of the dynamic similitude law is discussed.

1. Introduction

Scale model studies were performed to determinethe forces acting vertically on an emergency gate andits suspension. The only way of studying the behaviorof the emergency gate’s vibrations under extreme flowconditions was to use a physical scale model. Resultswere already available from several studies relating toflow under flat gates and flow induced vibrations ofsolid structures [2,7,8,10]. In the case under study,however, the flow contained an additional swirl com-

∗Corresponding author.

ponent, which was generated by the turbine. A pro-nounced swirl component was produced when the tur-bine was running at an increased speed (disconnectionfrom the system). Likewise, lock-up of the guide vanesystem risked a situation which could lead to extremeflow conditions. The forces acting on the emergencygate and its suspension contained both static and dy-namic components.

After computing these forces, we were faced withthe problem of transferring the model conditions to theprototype. For the static forces, transfer to the proto-type is possible by using Froude’s laws of similitude.The section under study was short enough to allow thefrictional forces to be ignores which is a prerequisitefor using Froude’s laws. The static forces resultingfrom the swirling flow can be transferred in the sameway. The transfer mechanisms of the dynamic forces,however, can only be dealt with by using the dynamiclaws of similitude. Different vibration mechanismswere found to develop as a function of the opening sizeof the emergency gate. Characteristic vibrations wereseen to occur for the small gate openings. As the gatewas lifted, vibrations generated by the pulsations of theturbine swirl became increasingly important. The goalof this study was to establish the vibration behavior ofthe gate for different openings.

2. Model set-up and measuring equipment

A transparent material (Plexiglas) was used to formthe model, while the turbine blade, the guide vanesystem, and the gate leaf were fabricated from brass(Fig. 1), which made it possible to study the cavita-tion swirl formed under certain flow conditions. Fig-ure 1 shows the suspension of the gate very clearly. InFig. 2.a, longitudinal section through the power unit isshown. The kinematics of the emergency gate wereregistered by means of a force and an acceleration trans-ducer. Figure 2 also shows the arrangement of the mea-suring transducers. We generally used transducers al-lowing the non-distorted registration of signals to fre-

Shock and Vibration 7 (2000) 3–13ISSN 1070-9622 / $8.00 2000, IOS Press. All rights reserved

4 A. Predin, R. Klasinc / Emergency gate vibration of the pipe-turbine model

Fig. 1. Kaplan turbine model – bulb unit.

quencies of at least 300 Hz. For the efficient acqui-sition and analysis of the measured data we used anA/D computer board (Burr Brown with Visual Designersoftware). This set-up had a scanning rate of 2 kHzper channel. The blocks of 8192 measured digits wereused for preparing the power spectra and the correlationanalyses.

The water flow over the top of the gate shows up asan additional problem at the position where the acceler-ation transducer was placed; so, the accelerometer wascovered with the plastic foil pipe to keep the electricalcontacts dry.

The dynamic response of the measurement systemwas determined before each measurement; meaning,the frequency and the system damping of gate’s instru-mentation system must be determined. The natural os-cillation of the gate, which is supported by the spindle,was excited by a force pulse. The strong damping ofthe natural oscillation of the gate is evident (Fig. 3)from the relatively large decrease on the magnitude ofthe oscillation. The system damping is determined bylogarithm decrement of two successive gate oscillationmagnitudes using the equation

δ = ln(Mi

Mi+1

)(1)

For the system damping, if it is operated in air, thedamping ratio is 0.172, and if it is operated in water the

ratio is 0.195. The natural frequencies are determinedusing FFT analyses of the oscillation data for the twocases. First, for the case when the gate was surroundedby air, and second, when the gate was surrounded bywater. In first case, the dominant natural oscillationfrequency of the gate is determined to be between 310and 320 Hertz, and in the second case it is between 30and 36 Hertz (Fig. 4). For the both cases, the frequencypeak is quite clearly evident. In both power spectrarecords, some other spectral frequencies are present,but their magnitudes are much smaller. Force and pres-sure transducers were calibrated for each trial. Theforce transducer was calibrated by weights and the pres-sure transducer was calibrated using a known pressure.Linear responses were achieved at both cases.

3. Measurement results

Measurements were performed at the model’s max-imal head difference (difference between upper andlower water level) at different gate openings. The flowrate, turbine runner speed, pressure at the draft tubebelow the gate, acting force, and acceleration at thegate in the vertical direction were measured. Fromforce-time history records (Fig. 5) different force vi-brations at different gate opening were evident. At thegate openings of up to 16.7% the force vibration withhigher frequency is evident. When the gate openingis increased, the force vibration frequency decreased.The average value of the acting force increases by en-larging the gate openings up to 83.3%, and then it de-creases after that. The reason for that situation is thatat the higher flow rates (larger gate openings) the swirlflow occurs at the draft tube. The extreme swirl flowis present at the cavitating turbine operating conditions(at maximal flow rates). With this situation, the spi-ral vortex rope (or vortex core) appears in a draft tubeand often causes violent pressure fluctuation, which iscalled the draft tube surge. That vortex core is identi-fied by pressure measurements and it is already presentat approximately a 28% gate opening. The vortex coreappears quickly in the flow in the draft tube, when theturbine model is operated at extreme operating condi-tions. At normal turbine operating regimes, the swirledflow is present at the draft tube, but not in such an ex-treme cavitating form. Pressure-time history records(Fig. 6) confirm the existing vortex core in the flow atthe draft tube; the pulsating pressure form is evident atthe larger gate openings. The pressure frequency in-creases from 3.2 Hz at a 28% gate opening to 4.64 Hz

A. Predin, R. Klasinc / Emergency gate vibration of the pipe-turbine model 5

Force transducer

Accelerationtransducer

Vibrating direction

Pressure transducer

Fig. 2. Transducer arrangement at tested model.

2,96 2,97 2,98 2,99 3,00 3,01 3,02 3,03 3,04

Gate natu ra l osci l la ti o n

t [s]

Acc [g]

In Ai r

a)

b) -0,8

-0,3

0,2

0,7

3,50 3,60 3,70 3,80 3,90

Gate na tu ral osci l la ti on

t [s]

Acc [g]

In wa te r

-5

-3

-1

1

3

5

Fig. 3. Time history records of the gate natural oscillation in air (a) and in water (b).

at on approximately 83% gate opening and than de-creases to 3.66 Hz at an approximately 94% gate open-ing. The reason for this frequency change is probablythe fact that in the vortex core there is more and moreair/water-exhalation, and the vortex friction increases.Two dominant force-frequency peak areas are shownin the frequency “waterfall diagram” (Fig. 7) and in-dicate a dependency of the amount of the gate open-ing. Dominant frequency peaks are located at two areas

between 27 and 54 Hz and between 297 and 324 Hz.These two areas approximately correspond to the nat-ural frequencies. At larger gate openings, the higherspectral frequencies are present in the power spectrarecord. These frequencies probably correspond to theflow turbulence at higher flow rates a cross the drafttube. It can be shown that the gate vibrates with naturalfrequencies at the smaller gate openings and with thefrequency that correspond to the vortex rope round-up


0,000

0,005

0,010

0,015

0,020

0,025

0,030

0 200 400 600 800 1000

Gate natural oscillation - Water

f [Hz]

0

0,001

0,002

0,003

0,004

0,005

0,006

0 200 400 600 800 1000

Gate natural oscillation - Air

f [Hz]a) b)

Fig. 4. Power spectra records of the gate natural vibration, determined in air (a) and in water (b).

frequencies at higher flow rates or at larger gate open-ings. This fact is also confirmed by correlation analy-ses between pressure and force, acceleration and force,and between the acceleration and pressure. Figure 8shows the correlation between force and flow pressure;a large correlation between the measured data of forceand pressure. At smaller gate openings the observablecorrelation between force and pressures is not evidentin the correlation record.

Below a 20% gate opening the small values of theacting force are evident. The gate vibrations at this area(A, Fig. 9) had a frequency near to the gate’s naturalfrequencies. In this operating area, the gate vibrationsare self excited gate vibrations. The average valuesof the acting force increase (Fig. 9) when the gate isopen up to 20%. In this area, the alternating part ofthe force is relatively small. The force frequency ishigher than the corresponding natural frequency, so itcan be concluded that this is the beginning of the tran-sition area of self excited gate vibrations and forcedvibrations. Gate openings up to 28% are in a transi-tional area where gate vibrations are both self excitedand force excited by the vortex rope. In the area desig-nated by (B) in Fig. 9, the alternating part of the actingforce increases further. In area (C), with up to 70%gate opening, the vortex rope is fully developed. Forthis area, an almost constant acting average force andconstant alternating force magnitude are characteristic.Moreover, in area (C) the force vibrating frequency isalso constant between 3.17 and 4.6 Hz. In the area ofgate opening between 60% and 83% the frequency isalmost constant, approximately 4.5 Hz. On that basis,one can conclude that the flow vortex frequency is al-most constant. By further increasing the gate openingfrom 70% and up to 95%, the average value of the act-

ing force decreases, but at the same time the alternatingpart of the force remains relatively large and does notdecrease until a gate opening of almost 100%. Theforce pulsating frequency in the area (D) decreases. Itcan be concluded that the vortex rope filled up with thewater steam, which is characteristic for the cavitatingoperating regime. The vortex rope rolled-up frequencydecrease could be caused by the increase of the vortexfriction in the flow in the draft tube.

4. Gate vibration mathematical model

As the measured results show, the gate vibrationscould be divided into the three types: self excited vi-brations; forced vibrations, caused by pressure oscil-lations in the draft tube [1] and at the combination ofthose two.

The self-excited gate vibrations can be expressedas one a degree-of-freedom system with the viscousdamping vibration equation

aF = my + ξy + ky (2)

where y is the displacement due to the vibration and itstime derivations, k is the “spring” rigidity, ξ is dampingcoefficient, m is the gate mass with the spindle andacceleration transducer mass, and aF is the alternatingpart of the hydraulics force acting on the gate in thevertical direction and is determined by

af = Fh − Fh,mean. (3)

In Eq. (3), Fh is the common hydraulics acting forceandFh,mean is the mean value or mean hydraulics forcethat is acting on the gate lip in the vertical direction.All needed parameters from equation (2) could be de-


2

2,5

3

3,5

4

0 0,5 1 1,5 2 2,5 3 3,5

Force - 2,8% Open

t [s]

F [N ]

12

13

14

15

16

0 0,5 1 1,5 2 2,5 3 3,5 4

Force - 5,6% Open

t [s]

F [N ]

30

32

34

36

38

0 0,5 1 1,5 2 2,5 3 3,5 4

Forc e - 16,7% Open

t [s]

F [N ]

25

30

35

40

45

0 0,5 1 1,5 2 2,5 3 3,5 4

Force - 27,8% Open

t [s]

F [N ]

30

35

40

45

50

0 0,5 1 1,5 2 2,5 3 3,5 4

Force - 38,9% Open

t [s]

F [N ]

25

30

35

40

45

50

0 0,5 1 1,5 2 2,5 3 3,5

Force - 72,2% Open

t [s]

F [N ]

15

20

25

30

35

0 0,5 1 1,5 2 2,5 3 3,5

Force - 83,3 Open

t [s]

F [N ]

0

5

10

15

20

25

0 0,5 1 1,5 2 2,5 3 3,5

Force - 94,4% Open

t [s]

F [N ]

Gat

e op

enin

g di

rect

ion

(inc

reas

ed c

apac

ity in

dra

ft tu

be d

irec

tion)

Fig. 5. Time history records of acting force measured at the gate suspension in vertical direction at different gate openings, from 2.8% up to94.4%.


-6000-5000-4000-3000-2000-1000

0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0

Gate ope nin g - 2,8%

t [s]

p [Pa]

-10000

-5000

0

5000

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0

Gate opening - 27,8%

t [s]

p [Pa]

-6000-4000

-20000

20004000

6000

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0

Gate opening - 94,4%

t [s]

p [Pa]

Fig. 6. Time history records of the flow pressures at different gate openings.

031

6293

124155

186217

248279

310341

372403

434465

496527

558589

2,80%

16,70%

38,90%

61,10%

83,30%

0

0,001

0,002

frequency [Hz]

Gate opening

Fig. 7. “Waterfall” power spectra record of acting force at different gate openings.

termined theoretically (m, k) or experimentally (aF , ξ)by well-known procedures. The unknown values ofthe alternating part of the hydraulics force aF and thedamping coefficient ξ could be theoretically determinedusing equations [8], as can the vertical gate vibration.The alternating hydraulics force is

aF = cFρgB∆H ′, (4)

where cF is the force coefficient, ρ is flow density, g isacceleration due to gravity, B is the gate lip area, and∆H ′ is the acceleration of the water column.

∆H ′ =−LgAD

Q, (5)

where L is the culvert or diffuser length,AD is the dif-fuser’s cross-sectional area, and Q is the time derivation

of the discharge equation.

Q = Aeff

√2g∆H, (6)

where ∆H is the local head difference across the gapand Aeff is the effective gape area.

Aeff = ϕδw, (7)

where ϕ is the flow loss coefficient (0.9− 0.8), δ is thetemporary gate opening and w is the gate width.

When gate vibrations are excited by close to a single-frequency harmonic excitation caused by vortex ropeat the draft tube, or by a pressure disturbance in thedraft tube [3], (at area B, C and D, Fig. 9). Governingequations [5,11] for motion of viscously damped in thegeneral form


2,255

2,260

2,265

2,270

2,275

2,280

0,0 0,2 0,4 0,6 0,8 1,0 t [s]

A

Force - Pressure Correlati on Gate opening - 83,3 %

1,055

1,060

1,065

1,070

1,075

1,080

1,085

1,090

0,0 0,2 0,4 0,6 0,8 1,0 t [s]

A

Force - Pressure Correlati on Gate opening - 2,8 %

Fig. 8. Correlation record between acting force and flow pressures in the draft tube.

0

10

20

30

40

50

60

0 25 50 75 100

Minimum

Average

Maximum

F [N]

Gate opening [%]

A

B

C

Cavitation

∆H max

D

Fig. 9. Maximal, minimal and average values of the acting force in dependency of gate opening.

y + 2ξωny + ω2y =F

msin(ωropet+ ψ), (8)

where ωn is the natural frequency, ξ is the dampingratio of natural oscillation, F is the excited force, andm is the added mass considering the gate vibration inthe water, ωrope is the frequency of the vortex rope, andψ is the phase difference between the excitation and theclosely sinusoidal excitation. The vortex rope rolled-upfrequency can be modeled using as a base formulationfrom [9], which is valid for the Francis turbine. Thenew, modified formulation is adopted for the Kaplanturbine and it takes the form

ωrope =2Γ

Dhe(n/nmax)0,4 , (9)

where Dh is the turbine runner hub diameter, n is therunner speed, nmax is the runaway runner speed, andΓ is the circulation that can be determined as the sumof the circulation in the axial direction Γax and, thecirculation Γp around the runner blade profile

Γ = Γax + Γp. (10)

The circulation in the axial direction (axial direction ofthe draft tube) is determined as

Γax =π2n

z(D2

t −D2h), (11)


15

25

35

0,0 20,0 40,0 60,0 80,0 100,0

Gate opening [%]

rope

Measured

Calculated

Fig. 10. Vortex rope frequency.

where z is the runner blade number andD t is the runner(tip) diameter. The circulation around the runner bladeprofile, considered Euler’s main equation for axial tur-bomachines, is determined as

Γp = g∆HtotπDh

z(12)

where ∆Htot is the total head across the turbine runner.Equation (10) results are in good agreement with the

measured data for the vortex rope frequency (Fig. 10).For higher gate openings there is a larger disagreementbetween the calculated and measured vortex rope fre-quency. This is caused by the fact that the measuredfrequency dropped rapidly while the vortex rope wasfilling up with water vapor and collapsing. With thesubstitution of Eq. (10) in to (9), good results for thecalculated gate vibrations are achieved. As an example,Fig. 11 shows the calculated results for force vibrationsat the gate in the vertical direction for a gate open-ing of 83.4% as compared to measured results. Thegood agreement of the vibration frequencies is evident,whereas the vibration magnitudes have less agreement.The reason the calculated and measured vibration mag-nitudes not agree could be because the other spectralfrequencies such as the runner speed frequency and therunner blade frequency and their higher harmonics arenot considered.

5. Parameter transfer from model to prototype

The static behavior of the gate could be transferredfrom model to the prototype conditions using Froude’slaw of similitude. The oscillating pressure distribution,

to the gate in the vertical direction increases the oscil-lating force acting normal to the gate lip. The right sideof Eq. (9), gives the force, whereωrope is the frequencyof vortex rope. All parameters on the right side of Eq.(9) depend on the fluid properties and the geometry ofthe gate or gate lip. For force and frequency transferwe can use the general equation [12],

F = f(V, ρ, µ, L) (13)

where V is flow velocity, ρ is flow density, µ is fluidviscosity, and L is the gate lip area minus the length.Using the Buckingham π theorems in equation (14)allows the equation to be rewritten as a relationshipbetween dimensionless parameters, such as Reynolds

Rem =ρmVmLm

µm=ρpVpLp

µp= Rep (14)

and Froude

Frm =V 2

m

gLm=V 2

p

gLp= Frp (15)

numbers. Using these dimensionless parameters, thetransfer between the model and prototype conditionscan be established. Considering the dynamic similari-ties that exist when the model and the prototype havethe same length-scale ratio, time-scale ratio, and force-scale (or mass-scale) ratio, shows that the dynamic sim-ilarity exists simultaneously with the kinematic simi-larity, if the model and the prototype force and pres-sure coefficients are identical. This is certain wherethere is free surface flow and when the model and theprototype have the same Reynolds, Froude, Weber, andEuler (cavitation) numbers.

The Froude number contains only length and timedimensions, and hence is a purely kinematic parameterwhich fixes the relation between length and time [6].So, from equation (16), if the length scale is

Lm = λLp, (16)

where λ is a dimensionless ratio, the velocity scale is

Vm

Vp=

(Lm

Lp

)1/2

=√λ. (17)

The time scale is

Tm

Tp=Lm/Vm

Lp/Vp=

√λ (18)

or the frequency scale is

ωm

ωp=Vm/Lm

Vp/Lp= λ−1/2 (19)


15

17

19

21

23

25

27

29

31

33

35

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0

Calculated

Measured

t [s]

F [N] Gate opening - 83 %

Fig. 11. Calculated and measured force comparison.

Gate

Prototype

Fp,p Fi,p

Ff,p

v

ξpkp

Ap

Gate

T

Model

T

km ξm

Am Fi,m

Fp,m

Ff,m

Fig. 12. Gate – model and prototype and kinematically similar flows.

using these equation the vortex rope frequency fromthe model to the prototype can be predicted. The exci-tation force, at the right hand side of equation (9), canbe predicted from the model to the prototype consider-ing Newton’s law, which says that the inertial force isthe sum of the pressure, gravity, and the friction force.Thus, the dynamic similarity laws listed above ensurethat each of these forces will be in the same ratio andhave equivalent directions between the model and theprototype. Let us examine two dynamically similar, in-compressible, viscous flows through the gates, a modelflow and a prototype flow (Fig. 12). Neglecting thebody forces, two types of forces acting on each flowparticle may be distinguished, namely, shear force andpressure force. If the inertia term in Newton’s lawis now written as a D’Alembert force, ma, we may

consider the sum of the two external forces plus theD’Alembert force as equal to zero and, thus, in equi-librium. The triangle of forces could be established foreach point of the flow as is shown in Fig. 12 for pointsAm andAp. According to the rules of dynamic similar-ity, the triangles of forces for these points are similar,since the sides of the triangles must be parallel. Wemay now form the following equations, which apply toall corresponding points:

(Press., F)m

(Press., F)p=

(Frict., F)m

(Frict., F)p(20)

=(Inert., F)m

(Inert., F)p= const.

The following relations may be derived from theseequations:


(Inert., F)m

(Frict., F)m=

(Inert., F)p

(Frict., F)p= (const.)1, (21)

(Inert., F)m

(Press., F)m=

(Inert., F)p

(Press., F)p= (const.)2, (22)

Using derivations [4], the end equation (20) can bewritten as(

∆pρV 2

)m

=(

∆pρV 2

)p

(23)

where ∆p is the difference between the static pressureat the draft tube entrance and the static pressure underthe gate, and V is the entrance flow velocity at the drafttube entrance. From Eq. (23), it may be concludedthat a necessary condition for dynamic similarity forthe particular flows under study is the equality of theReynolds number and the Euler number between theflows, using the same flow parameters and geometricalscale. In our case, we can replace ∆p in equation(23) with F/(Bw), which is proportional to ∆p (B isthe gate lip area and w is the gate width). Using thissubstitution, we can write

Fm/(Bmwm

ρmV 2m

=Fp/(Bpwp

ρpV 2p

(24)

hence the force on the prototype is

Fp = Fm

ρpV2p

ρmV 2m

(Bpwp

Bmwm

). (25)

Assuming a ratio λ, from geometrical similarity, equa-tion (17), between the model and the prototype, equa-tion (26) can be rewritten as

Fp = Fmρp

ρmλ3 (26)

which applies to the incompressible flow, where ρp =ρm = ρ = constant, i.e., Froude’s law.

The gate guiding system rigidity or the “spring”rigidity k of the model and the prototype and the gatemass, respectively, could be in the same geometric ratio

kp

km=mp

mm= λ. (27)

If so, then it is acceptable that the damping coefficientsare also the in the same ratio

ξpξm

= λ. (28)

If the stiffness, mass, and damping coefficients do nothave the same ratio, then we must measure or calculatethem directly from the prototype. Allowing for theabove equations (26) and (29), the force vibration atthe prototype emergency gate can be predicted fromcalculations using equation (9).

6. Conclusions

The emergency gate vibrations that occur by closingthe draft tube of the Kaplan horizontal shaft turbineare generated as flow-excited vibrations. In the area ofsmall gate openings, about up to 20%, the vibrationsare self excited. The common vibrating frequency isclose to the gate’s natural frequency which means thatthe gate vibrates in the vertical direction with its naturalfrequency. Fortunately, the magnitude of these vibra-tions is small, up to 20% of hydrodynamic force thatof the. With larger gate openings, from 25% to 70%, avortex flow in the draft tube is created. Vortex excitedgate vibrations are generated simultaneously which arealmost sinusoidal. The common vibration frequencyis equal to the flow pressure pulsation frequency. Thepressure pulsation in the draft tube is a consequenceof the vortex rope created in the flow. The magnitudeof the pulsation is up to 50% of the average hydraulicsforce at these openings. With maximal gate openings(from 75% to 100%) the vibration magnitudes increaseintensively and are up to 70% of the average hydro-dynamics force at these openings. However, the av-erage value of the hydraulics force at these openingsdecreases to very small values when the gate is fullyopen.

Finally, it can be concluded that the critical momentoccurs when the extreme gate loading exists, that is,at the first quarter of the gate closing from 100% to75% of the gate opening when the maximal flow ratein the draft tube is achieved. In this operating areathe cavitating operating conditions at the turbine runnerand in the turbine draft tube show up. That means thatthe vortex from the turbine runner is changed into acavitating vortex rope that is rolled-up at the draft tube.

Transferring parameters from the model to the proto-type is possible by allowing for dynamic similarity, theFroude law, and geometrical similarity. On that basis,it is possible to predict force vibrations at the prototype.

Acknowledgments

The authors thank the Institute of Water Buildingsand Water Sciences at Graz Technology University inAustria for their support of these investigations as wellas for permission to publish the results.


References

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[2] D.R. Blevins, Flow-Induced Vibration, second edition, VanNostrand Reinhold, New York, 1990.

[3] H. Brekke, Ongoing and planned work at the Norwegian In-stitute of Technology, related to hydraulic machinery understeady oscillatory conditions, 7th IAHA meeting, WG1, Ljubl-jana 1995.

[4] V.R. Giles, J.B. Evett and C. Liu, Schaum’s outline of: Theoryand problems of Fluid mechanics and hydraulics, McGraw-Hill Book Company, 1994.

[5] P.J. Holman and R.J. Lloyd, Fundamentals of Mechanical Vi-brations, McGraw-Hill Book Company, 1993.

[6] T. Jacob, Similitudes in stability of operation test for Francis

turbines, Hydropower & Dams, Jan 1994.[7] R. Klasinc and A. Predin, Vibrations of an Emergency Gate in

the Draft Tube of a Horizontal – Shaft Turbine, Proceedingsof the 28th IAHR congress, Graz, 1999.

[8] P.A. Kolkman, Flow-induced gate vibrations, thesis work,1976, Delft Hydraulics Laboratory, publication no. 164.

[9] M. Nishi, X. Wang and T. Takahashi, A short Note on theRotating Frequency of Vortex Rope in a Draft Tube, 7th IAHRMeeting, WG1, Ljubljana 1995.

[10] K. Ogihara, Y. Ueda, I. Minagawa and T. Ueda, Fields test ofvibration of draft gate in pump storage station, Proceedings ofModeling, Testing and Monitoring for Hydropower Plants-IIIConference, Aixen-Provence, France, October 1998.

[11] A. Predin, Torsional Vibrations at Guide-Vane Shaft of Pump-Turbine Model, Shock & Vibration, Volume 4, Issue 3, 1997,153–162.

[12] M.F. White, Fluid Mechanics, second edition, McGraw-HillBook Company, 1986.

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Emergency gate vibration of the pipe-turbine modeldownloads.hindawi.com/journals/sv/2000/406976.pdf4...

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Transcript of Emergency gate vibration of the pipe-turbine modeldownloads.hindawi.com/journals/sv/2000/406976.pdf4...