correlation of acoustic fluid-structural interaction...

Correlation of Acoustic Fluid-Structural Interaction Method for Modal Analysis with Experimental Results of a Hydraulic Prototype Turbine Runner in Water

B. Graf1, L. Chen2 1 Voith Hydro Holding GmbH & Co. KG Alexanderstraße 11, Heidenheim, Germany email: [email protected] 2 Voith Hydro, Inc 760 East Berlin Rd., York, PA 17408, USA email: [email protected]

Abstract A lot of runner damage occurs in last years in hydro industry, which is more or less related to over pursue the reduction of runner thickness/weight ratio to save material cost and improve hydraulic performances, and over trust the reliability of structural analysis with finite element method. It is the first important step to more accurately predict runner’s frequencies and mode shapes in water and then to modify the runner design to weaken the runner blade - wicket gate leaf interactions, i.e. rotor - stator interactions (RSI), to as less as possible.

Acoustical fluid-structural interaction (AFSI) method has been adopted in hydraulic industry for runner dynamic response analysis in water for years. To evaluate the reliability of AFSI method, an experimental modal analysis (EMA) of a hydraulic full size prototype Francis runner with three meter of diameter is made in air and immerged in a cubic water pool. Finite element analysis (FEA) with AFSI calculation has been performed with different boundary conditions and compared with the EMA. We pay attention to both natural frequencies and mode shapes of whole runner and blades. A great agreement between numerical and experimental results has been achieved, and demonstrated that EMA as well as AFSI methods in predicting runner dynamic behaviour is sufficient accuracy. Hence an experimental measurement of the natural frequencies and mode shapes in air is sufficient to get a conclusion on the accuracy of the calculated modal parameter in water.

Further study of runner dynamic behaviour in water as in operation environment has been performed. The comparison of the runner frequencies and mode shapes in water between as in measured and as in operation showed the impact of water volume change on runner dynamic behaviours, especially for lower modes, e.g. one nodal diameter k1 and two nodal diameters k2 modes

1 Introduction

It is well known that static behavior study of the runner is quite reliable and robust with today's advanced FEA and Computational Fluid Dynamic (CFD) simulation software. However, dynamic behavior, including natural frequencies, mode shapes, characteristic damping and response amplitudes of the runner submerged in water in operation site is still a key issue and challenge to hydraulic and mechanical design and rehab in hydro industry today. It is essential to more accurately predict runner’s frequencies and mode shapes and then to modify the runner design to weaken the runner blade - wicket gate leaf interactions, i.e. rotor - stator interactions (RSI), to as less as possible.

Years ago in hydro-turbine industry, traditionally the modal analysis of a runner was performed numerically in air, and then all the natural frequencies of the runner in water were determined

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approximately by a constant reduction factor, say 0.7, applied to the ones in air. Basically in this kind of estimation the reduction factor is independent on runner geometries and mode shapes.

It is well known that the frequency reduction factor from air to water is not a constant, and really dependent on the types of the runner, runner geometries and the mode shapes. Acoustical fluid-structural interaction (AFSI) method has been adopted in hydro-turbine industry for runner dynamic response analysis in water for years. So far there is only few reference [9] available about the accuracy study of the method.

A comprehensive experimental investigation of a full size prototype Francis runner has been done in a cubic water pool using experimental modal analysis (EMA) method in Voith Hydro. The runner was put upside down on the bottom of the pool, and supported by three soft wood/rubber, as shown in figure 1. Hence the impact of the support on whole runner’s vibration is limited, but nevetheless considered in this investigation.

Figure 1: A schematic overview of the entire runner and water pool and runner support

An exact water pool and the prototype Francis runner set as in measured has been numerically simulated by AFSI with optimized boundary conditions, as shown in Figure 2. A great agreement between measured and numerical results has been achieved, and demonstrated that AFSI method in predicting runner dynamic behavior is sufficient reliable.

Further modal analysis of runner in water as in operation has been performed. The comparison of the runner frequencies and mode shapes in water as in measured and as in operation showed the significant impact of water volume change on runner dynamic behaviors, especially for lower modes, i.e. one nodal diameter k1 and two nodal diameters k2 modes.

flange Wood/rubber supports

Runner band

Wood/rubber supports

wood

rubber

2490 PROCEEDINGS OF ISMA2010 INCLUDING USD2010

Figure 2: A schematic overview of the numerical model of entire runner and water pool

2 Theoretical background

2.1 Acoustical fluid-structural interaction

In acoustical fluid-structure interaction problems, basically we have to study simultaneously

• structural behavior in structural domain;

• acoustical fluid behavior in fluid domain,

• acoustics fluid-structure coupling behavior at the fluid-structural interface.

The structural behavior in structural domain is governed by structural dynamics equation, discretized as [1]

[ ] [ ] [ ] eeeeeee FuKuCuM =++ &&& (1)

Where [Me] is the structural mass matrix, [Ce] structural damping matrix, [Ke] structural stiffness matrix, eu&& nodal acceleration vector, eu& nodal velocity vector, eu nodal displacement vector and

eF applied load vector.

Acoustical fluid behavior in fluid domain is governed by the acoustic wave equation, determined by the fluid momentum (Navier-Stokes) and continuity equations as [8]

01 2

2

2=∇−

∂∂

Pt

P

c (2)

Where c is the speed of sound 0/ ρk in fluid medium, 0ρ mean fluid density, k bulk modulus of fluid,

P acoustic pressure and t time.

Following assumptions have been used for equation 2,

• The fluid is compressible (density changes due to pressure variations).;

• The fluid is inviscid (no viscous dissipation);

• There is no mean flow of the fluid;

MODAL TESTING: METHODS AND CASE STUDIES 2491

• The mean density and pressure are uniform throughout the fluid.

The lossless wave equation 2 can be discretized and written in matrix notation with taking coupling mass matrix due to fluid-structural interface into account, as [2; 14]

[ ] [ ] [ ] 00 =++ eT

eePee

Pe uRPKPM &&&& ρ (3)

Where [ ]PeM is the fluid mass matrix, [ ]eP nodal pressure vector, [ ]eu nodal displacement component

vectors, [ ]PeK fluid stiffness matrix, [ ]eR0ρ coupling mass matrix (fluid-structural interface).

By accounting for the fluid pressure load acting at the structure-fluid interface, and for the energy loss due to damping at the absorbing boundary surface, the structural equation (1) and wave equation (3) can be rewritten as

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

=++

+

=−++

00

.

eT

eep

eePee

Pe

eeeeeeeee

uRPKPCPM

FPRuKuCuM

&&&&

&&&

ρ (4)

Where [ ]peC is fluid damping matrix. Equation 4 describes the complete finite element discretized

equations for the fluid-structure interaction problem.

2.2 Fundamentals of the Experimental Modal Analysis

The analytical Modal Analysis as well as the Experimental Modal Analysis (EMA) is for the extraction of the Modal Parameters of a structure. In contrast to the FE-calculation, which is carrying through a eigenvalue and eigenvector analysis, the EMA uses a Curve-Fitting-Procedure to determine the coefficients of functions fitting best with the measured spectrums [4; 6]. For complex structures with almost linear behaviors and close situated natural frequencies Multiple-Degree-of-Freedom-Methods (MDOF) are adopted and we are able to use these methods to consider several references j (jmax = J) and several hundred response coordinates k (kmax = K). Such MIMO-techniques (Multiple Input Multiple Output), as the Frequency Polyreference Residue Technique or the PolyMax-Technique described herein, are working in the frequency domain and consider the residues of the modes outside the analysed frequency band. For the extraction of all modal parameters the following steps are necessary:

• Installation of the test setup

• Broad-band excitation of the structure

• Measurement of the reference and response time signals fj(t) and xk(t)

• Multiplication of the time signals with window functions w(t)

• Transformation into the frequency domain:

∫+∞

∞−

−= dtetwtxX tikk

ωω )()()(

• Calculation of the Frequency Response Functions (FRF):

)(

)()(

ωω

ωk

jjk F

XH =

• Determination of the modal parameters for each FRF

• Selection of the modal parameters with the stability diagram

• Calculation of all modal parameters of the structure


2.2.1 Determination of the modal parameters

After the calculation of the Frequency Response Function between the K references and the J responses, the determination of modal damping, natural frequencies and vectors for each FRF is the next step. These parameters are all inherent of the structure and therefore should be the same for each FRF. For the

extraction of the modal parameters from the experimental Frequency Response Function Hjk( Ω ) the same analytical approach as for analytical Forced Response Analysis is used. The selected frequency band for

analysing Ω min ... Ω max contains the natural frequencies m1 ... m2. The influence of the natural frequencies situated unavoidable above the upper and below the lower limit of the selected frequency band are considered with the residual mass MR

jk and the residual stiffness KRjk in equation (5):

Rjk

m

mr rrr

jkr

Rjk

jk Kii

A

MH

1

2

1)(

~ 2

1

222+

Ω−Ω++

Ω−=Ω ∑

= ξωω (5)

By determination of the square sum error (equation 6)

( )2

max

min

)(~

)(∑Ω

Ω=Ω

Ω−Ω= jkjkjk HHE (6)

and a partial differentiation with respect to the unknowns q,

Rjk

Rjkjkjk

jk MKAAqq

E,,...,,,...,,,...,,;0 212121 ηηωω==

∂∂

. (7)

the unknown parameters can be calculated with iterative, non-linear least-square techniques.

2.2.2 Selection of the modal parameters

The number of natural frequencies and thus the number of the poles m1-m2+1 to calculate is not known during the determination of the modal parameter (described in chapter 2.2.1). To determine the number of poles, their number is increased stepwise. As the parameters are stable over all FRF they are marked in a stability diagram (figure 3).

Figure 3: Stability Diagram

Frequency


Poles can be regarded as numeric if they do not converge with a increasing number of used poles [3]. In literature it is recommended [7] that the maximum number of assumed poles should be three times higher than the number of expected experimental modes. To distinguish between physical and numerical modes it is recommended to use the Mode Indicator Function (MIF) and/or the Modal Confidence Factor (MCF) [4; 11]. The MCF is applied to evaluate repeated phase-conditions which are fulfilled by physical modes but meaningless for numerical. The MCF is one for physical and zero for numerical modes [12].

The MIF is calculated with the following formula:

∑∑=

²||

|]|)([Re

H

HHalMIF

. (8)

This function has a maximum value of one. Each minimum is indicating a natural frequency [3].

With the natural frequencies, modal damping and vectors selected in the analysed frequency band it is possible to calculate the mode shapes with a Polyreference Technique.

2.3 Calculation of all modal parameters

The Polyreference Residue Technique described herein considers response functions of several references to receive the global parameters by least-square technique.

As the most MIMO - techniques working in the frequency domain, this technique is also based on the impulse-response function. The impulse-response function describes the dynamic response at the location j as a reaction of a normalized excitation at the location k [4] and is developed by Vold [13] in the time domain:

ti

jk

ti

jkjkr

rrrr

eAeAth

−−−

−+−

+=22 1

2

1

1)(ξωξωξωξω

(9)

ts

jkts

jkjkrrr eAeAth ⋅⋅ ∗

+= 21)( with

21 ξωξω −+−= rrr is. (10)

The impulse-response function of the MDOF-system is received by superposing the N Modes.

( )∑

=

⋅⋅ ∗

+=N

r

tsjk

tsjkjk

rr eAeAth1

21)( (11)

7,5,3,1)( 11

2

1

==== ∗+

∗+

=

⋅∑ rssAAeAth rrrr

N

r

tsjkrjk

r

(12)

Transforming the response-function into the frequency domain, by applying the Fourier-Transformation [11], leads to the complex Transfer Function (13):

∑

= −Ω=Ω

N

r r

jkrjk si

AH

2

1

)( (13)

The equations of each of the L frequency lines within the frequency band Ω max – Ω min form the system of equation 14.


=

Ω

Ω

Ω

−Ω−Ω−Ω

−Ω−Ω−Ω

−Ω−Ω−Ω

jkN

jkr

jk

sisisi

sisisi

sisisi

jk

jk

jk

A

A

A

H

H

H

Nr

Nr

Nr

2

1

111

111

111

max

min

2maxmax1max

21

2min´min1min

)(

)(

)(

M

M

LL

MOMM

LL

MMOM

LL

M

M

(14)

By considering several response-functions (SIMO – Single Input Multiple Output) it is possible to expand equation 15:

JNJkNjkNkN

Jkrjkrkr

Jkjkk

NNsisisi

sisisi

sisisi

JLJkjkk

Jkjkk

Jkjkk

AAA

AAA

AAA

HHH

HHH

HHH

Nr

Nr

Nr

××−Ω−Ω−Ω

−Ω−Ω−Ω

−Ω−Ω−Ω

×

=

ΩΩΩ

ΩΩΩ

ΩΩΩ

22212

1

1111

22

111

111

111

maxmaxmax1

1

minminmin1

2maxmax1max

21

2min´min1min

)()()(

)()()(

)()()(

LL

MOMM

LL

MMOM

LL

LL

MOMM

LL

MMOM

LL

LL

MOMM

LL

MMOM

LL

(15)

Up to now, only one reference is considered. To expand the SIMO System of equation (15) to a MIMO

Technique the values of the Frequency Response Function Hjk( Ω ) have to be regarded as K-dimensional column-matrices. Hereby every value of the matrix is correlated to a excitation point k. To take this expansion into account on the right side, it has to be considered, that the mode shapes are excited at every excitation point with a different intensity. The shape keeps the same for every excitation point, and only the amplitudes are scaled different Wr k = Ar k / Ar kref for each reference Ar kref. Thus the relationships between the vector components are identical. Each line in equation 15 (red dashed box) has to be replaced by the following system of equation to include several excitation points (references):

JNJNjNN

Jrjrr

Jj

NNsi

si

si

NKKNKrK

kNkrk

Nr

JKJKjKK

Jkjkk

Jj

AAA

AAA

AAA

WWW

WWW

WWW

HHH

HHH

HHH

N

r

××−Ω

−Ω

−Ω

××

=

ΩΩΩ

ΩΩΩ

ΩΩΩ

22212

1

1111

22

1

1

1

221

21

12111

1

1

1111

2

1

00

00

00

)()()(

)()()(

)()()(

LL

MOMM

LL

MMOM

LL

LL

MOMM

LL

MMOM

LL

LL

MOMM

LL

MMOM

LL

LL

MOMM

LL

MMOM

LL

(16)

This equation can be written in a short form:

[ ] [ ] [ ][ ][ ]

JNjkNNNKJK AUWH×××× Ω=Ω

2222 )()(, (17)

Hereby [W] is the correlation-matrix. This matrix contains the relations of the summed up FRFs referred to the references.

To consider the residues at the boarders of the selected frequency bands, the matrices [W], [U] and [A] have to be expanded to [W]*, [Z]* and [A]* (equations 18 - 20):


(18)

[ ] [ ] [ ] [ ][ ][ ]

[ ][ ]

[ ][ ][ ]

JKNJK

R

JKR

JN

kNKN

KNKKKKKNKJK

K

M

A

Z

Y

U

IIWH

×+×

×

×

+×+

+×××××

ΩΩ

=Ω

22

2

2222

222 )(

)(

)(

(19)

[ ] [ ] [ ][ ][ ]*

22222222 )()(JKNjkKNKNKNKJK AUWH

×+∗

+×+∗

+×× Ω=Ω, (20)

The insertion of residue terms in equation 15 leads finally to the system of equation to solve.

[ ]

[ ]

[ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ]∗

∗∗

∗∗

∗∗

×

×

×

Ω

Ω

Ω

=

Ω

Ω

Ω

jk

JK

JK

JK

A

UW

UW

UW

H

H

H

)(

)(

)(

)(

)(

)(

max

min

max

min

M

M

M

M

. (21)

The mode shapes can be finally calculated by using the least square sum (equation 22 - 24).

[ ] [ ][ ]APH = (22)

[ ] [ ] [ ] [ ][ ][ ]APPHP TT = (23)

[ ] [ ] [ ][ ] [ ] [ ][ ]HPPPA TT 1−= (24)

With weak damping and linear structure behaviour, the real modes can be assumed. Thus the imaginary part of the mode shapes Ajk is neglectable.

[ ]

KNKKNKrK

kNkrk

Nr

JK

WWW

WWW

WWW

H

2221

21

12111

100

010

001

100

010

001

)(

+×

×

=Ω

LL

MOMM

LL

MMOM

LL

LL

MOMM

LL

MMOM

LL

LL

MOMM

LL

MMOM

LL

kNKN

si

si

N

2222

1

1

1

1

10

01

0

0

0

0

2

2

2

1

+×+

Ω−

Ω−

−Ω

−Ω

L

MOM

L

L

MOM

L

L

MOM

L

JKNKK

KK

MM

MM

JNN

J

RJK

RK

RJ

R

RJK

RK

RJ

R

AA

AA

×+

22

11

11

11

11

212

111

1

111

1

111

L

MOM

L

L

MOM

L

L

MOM

L


3 Experimental Modal Analysis

For the identification of the experimental modal parameters, the transfer functions between up to 15 reference sensors and the response points are measured. Therefore the runner was excited with an impulse hammer (excitation in air) and a force cell with rod (excitation in water). Afterwards the natural frequencies were determined by using a stability diagram and a Polyreference Residue Technique [10].

Response points are the locations excited by the impulse hammer. At each trailing edge the 9 response points are selected. At the band the 4 response points are selected near the leading edges. Totally 195 response points were excited with the impulse hammer.

Reference sensors were glued on the trailing edges and on the band near the leading edge of the runner

The prototype Francis runner was put upside down on the bottom of a water pool of a volume greater than 50 m³, see figure 3. The runner was put on three wood/rubber supports. Hence, the support is quite soft and has not so much influence on the vibration of the runner.

Figure 3: Experimental Modal Analysis at the runner in water and k = 2 mode

4 Numerical model

Exactly the same prototype Francis runner set in the water pool as in measured, shown in Figure 1 was numerically simulated, exhibited in Figure 4, as

• The movement of the Francis runner, water and runner-water interface was governed by Equation (4).

• The six outside surfaces of the water pool were properly simulated as without displacement movement.

• In the simulations the wood/rubber supports (WRS) were modeled to keep the simulation comparable to the measurement.


Please note that the combined runner-water model of the Francis runner does not constitute a cyclic symmetric structure due to the cubic water pool. Hence a complete 3600 finite element model, including details such as blade fillets, of the coupled water-runner system was adopted to capture the dynamic characteristics. The finite element model shown below in Figure 4 consists of about 360,000 elements of the runner in the water pool, see details in table 1. Both inside and outside the runner was modeled by water acoustic elements.

Figure 4: FEA model for modal analysis in a water pool

Solid element # Fluid element # Node # Active DOFs Software

120,000 240,000 240,000 850,000 (ANSYS 11.0)

Density – ρ = 1,000 kg/m3

Young’s modulus - E = 2.1E5 MPa

Sound speed - c = 1100 m/s

Runner properties: Poisson’s ratio – υ = 0.3Density – ρ = 7830 kg/m3

Fluid properties:

Table1: Details of the FEA model for modal analyses in water pool

5 Comparison between numerical and experimental analysis

Figure 5 - Numerical and experimental mode shapes of zero order b0 with five nodal diameters k5


Figure 5 shows a comparison of the results of the numerical and the experimental modal analysis. On the left side is displayed the mode shape, calculated by a finite-element-analysis (FEA), with five nodal diameters k5. In radial direction there is no node line along the trailing edges, hence it is a zero order mode b0. At the right side of figure 5 is shown the mode shape measured by experimental modal analysis (EMA). At the trailing edges the nodal diameters are clearly visible in air as well as in water. At the band the node lines are also present but often superposed by vibrations of different wave lengths.

Figure 6 shows the natural frequencies of the numerical and experimental mode shapes. The values are drawn up according to the number of tangential node lines k (nodal diameters) at the band or crown [5]. In this figure are only displayed the modes with zero radial node lines.

Figure 6: Comparison of the measured and the calculated frequencies with zero order modes b0 in air and in water

The mode with one node line k1 tilts around the wood/rubber support and is dominated by both boundary condition and the runner. It is a mode with a low frequency and not considered in the diagram.

Between measured and calculated natural frequencies the agreement is great for hydro-turbine industry’s practice and acceptance. The maximum deviation in air is less than -3.4 % (average 2.4 %) even without updating the global material parameters. In water the deviation of the measured and calculated zero order modes are between 0.3 % and 6.3 % (average 3.4 %).

The increased complexity of the first order modes b1 does not influence the accuracy of the finite-element- and the acoustical fluid-structure interaction-calculations as apparent in figure 7. The maximum deviation in air is less than -2.0 % (average 1.0 %). In water the deviation of the measured and calculated zero order modes are between -4.5 % and 6.7 % (average 3.1 %).

The correlation between the numerical and analytic natural frequencies in air and water leads to the conclusion, that for hydraulic runners it is sufficient to prove that the modal parameters in air are within a tolerable limit after the manufacturing in the shop.

k2 - two tangential node lines (nodal diameters)

b0 - zero radial node lines


Figure 7: Comparison of the measured and calculated first order modes b1 in air and in water

6 Comparison of numerical analysis in water as in measured and as in operation

In order to study the impact of the water environment surrounded the runner on the frequencies and mode shapes, the water in the cubic pool, as shown in Figure 1, was replaced by the water volume as in operation, as shown in Figure 8.

Figure 8: FEA model for modal analysis in water as in operation

Herein the details of water at the gaps and chambers among the head cover, seals, draft tube, crown and band are modeled as accurately as possible. Impedance boundary at inlet and outlet of the runner was used to simulate the fluid boundary condition of non-reflecting. Please note that in this investigation we only

k3 - three tangential node lines (nodal diameters)

b0 - one radial node line


pay attention to the water volume change, including its boundary condition, and the influence of shaft on modes k0 and k1 is not included, i.e. free-free boundary condition employed.

The comparison of numerical analysis of the runner in water as in measured and as in operation (not including shaft’s influence) is reported in figure 9 for zero and first order modes. Herein k0 with zero number of node lines in tangential direction represents every blade bending in the same way, and blade bending deformation is dominant, see figure 9 (right side).

Figure 9: Comparison of water volume as in measured and as in operation

Figure 9 demonstrates that

• There exists a significant impact of runner water environment on low k mode frequencies, e.g. k1 and k2 modes, and the max deviation is about 10% to 20%. This is mainly valid for zero order modes. First order modes are less sensitive to the water volume, due to their more complex mode shapes.

• There is only a limited influence of runner water change on high mode frequencies, i.e. nodal diameters greater than 2, and the difference is less than 4%. Basically compared to the water as in measured in the pool, the volume around the runner in the operation is confined by the space outside of the runner. However, in the operation environment water comes all the way from spiral case, and basically the water surfaces at entrance are free; and in measurement, the water boundary is fixed at five water surfaces without any movement except the top one. Hence the submerged runner in the pool behaves more “stiff” than as in operation, and its frequencies are higher except the k1-b0 and the k0-b1 modes.

• There is only a limited influence of runner water volume change on the mode with zero tangential node lines k0, and the difference is about 4%;

• A mode with zero tangential node lines occurs at operation condition, but not at the water pool.

• Even for the same k mode, there is not only the difference of frequency, but also the details of runner deformation, as shown in figure 10, of the blade, crown and band.

k0 - zero tangential node lines (nodal diameters)

b0 - zero radial node lines


Figure 10:- k = 5 mode shapes as in measured and as in operation

7 Conclusion

The dynamic behaviors of Francis runner’s mode shapes and frequencies in water can be predicted numerically by the methodology of acoustical fluid-structure interaction and can be measured by the experimental modal analysis. The comparison of the results shows, that they are reliable and acceptable for hydro-turbine industry to get inside of runner dynamic response during operation.

There is a significant impact of water environment surrounded the runner on both mode shapes and frequencies, especially for modes up to two tangential node lines. Due to water environment change, some mode shapes may disappear. Hence a high attention should be paid to simulate the water surrounded the runner as accurately as possible.

If an experimental check of the natural frequencies and mode shapes in air demonstrates an acceptable correlation with the numerically simulated ones, the modal parameters in water numerically calculated should be reliable with an acceptable exactitude as well. Hence an experimetal modal analysis in air is sufficient to get a conclusion on the exactitude of the calculated modal parameter in water too.

Acknowledgements

The authors would like to acknowledge VOITH Hydro St. Pölten for their collaboration.

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[7] D. L. Hunt, R. Brillhart, Practical application of the modal confidence factor to the polyreference method. IMAC 4. 1986

[8] E. L. Kinsler, et. al., Fundamentals of Acoustics, John Wiley and Sons, New York pp. 98-123 (1982)

[9] Q.W. Liang, et al, Modal Response of Hydraulic Turbine Runners, 23rd IAHR Symposium, Yokohama, 2006

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[13] H. Vold, G. T. Rocklin, The Numerical Implementation of a Multi-Input estimation Method for Mini-Computers. First International Modal Analysis conference. 1982

[14] O. C. Zienkiewicz, R. E. Newton, Coupled Vibrations of a Structure Submerged in a Compressible Flui" , Proceedings of the Symposium on Finite Element Techniques, University of Stuttgart, Germany (June 1969)


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