ML17264B052
| ML17264B052 | |
| Person / Time | |
|---|---|
| Site: | Ginna  |
| Issue date: | 09/18/1997 |
| From: | Vissing G NRC (Affiliation Not Assigned) |
| To: | NRC (Affiliation Not Assigned) |
| References | |
| TAC-M95759, NUDOCS 9710060300 | |
| Download: ML17264B052 (29) | |
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UNITED STATES NUCLEAR REGULATORY COMMISSION WASHINGTON, D.C. 205554001 September 18, 1997 LICENSEE:
Rochester Gas and Electric Corporation FACILITY:
R.
E. Ginna Nuclear Power Plant
SUBJECT:
SUMMARY
OF MEETING WITH REPRESENTATIVES OF ROCHESTER GAS AND ELECTRIC CORPORATION ON AUGUST 25,
- 1997, CONCERNING STRUCTURAL CONSIDERATIONS FOR THE PROPOSED RERACKING OF THE SPENT FUEL STORAGE POOL (TAC NO. M95759)
Representatives of Rochester Gas and Electric Corporation (RGS,E) together with representatives of the RG&E vendors, Framatome Cogema Fuels (FCF) and Framatome Societe Atlantique De Techniques Advancees (ATEA) met with the NRC staff (primarily the members of the Civil Engineering and Geosciences Branch) in the NRC offices in Rockville, Maryland, on August 25, 1997, to discuss the structural analysis of the proposed Spent Fuel Storage Pool (SFSP) modifications along with the justification for the use of the ANSYS code in the structural analysis.
Enclosure 1 provides the list of the attendees.
Enclosures 2 and 3 were used to support Framatome's discussion on the analysis of the proposed racks for the SFSP.
Enclosure 4 provides basic layout drawings of the spent fuel pool.
One of the purposes of the meeting was to discuss the justification for the use of the ANSYS computer code in the structural analysis of the proposed spent fuel racks.
The staff explained a concern regarding the ANSYS computer code.
Holtec International has submitted a 10 CFR 21 notification that alleges that disparate results for calculating displacements are obtained by the use of the ANSYS code depending upon the particular computer system used.
Also, they allege that disparate results for forces and displacements are obtained by the ANSYS code depending upon the technique employed to launch the analyses runs.
Framatome indicated that they used version 5.2 of the ANSYS code.
Using the material in Enclosure 2,
Framatome indicated that they have several test cases in which the hand calculations verified the computer results and they were satisfied with all the results.
The staff identified and discussed several issues related to the following that needed further explanation:
l.
The single safe shutdown earthquake (SSE) artificial time history used for stress analysis.
2.
The dynamic fluid-structure interaction analysis using the computer
- code, ANSYS.
3.
The dynamic fluid coupling element (FLUID38 of the ANSYS code) used in the analysis.
4.
The analytical simulation of the rattling fuel assembly impacting against the cell.
97'i00b0300 9709i8 IIIIIIIIIIIIIIIIIIIIIII.IIIIII
September 18, 1997 5.'tie deformation shape and magnitude of the deformations of the rack for the single-rack SSE analysis when the maximum displacement at the rack top corner occurs.
6.
The magnitude of the hydrodynamic pressure distribution along the height of the rack during the fluid and rack interaction for 3-D single and multi-rack analyses.
7.
Peak response results and stresses of the single and multi-rack SSE analyses.
8.
Impacts between a rack and concrete spent fuel pool wall and between racks.
9.
The quality assurance program to assure proper construction and installation.
The staff indicated that a more detailed request for additional information will be formally submitted following the meeting.
The licensee and its vendors representatives appeared to understand the concerns of the staff and indicated that they could respond to the formal questions within 30 days of receipt.
Sincerely, ORIGINAL SIGNED BY:
Docket No. 50-244 Guy S. Vissing, Senior Project Manager Project Directorate I-I Division of Reactor Pr'ojects - I/II Office of Nuclear Reactor Regulation
Enclosures:
1.
List of Attendees 2.
Results of the ANSYS code 3.
Technical paper on Dynamic Fluid-Structure Coupling of Rectangular Modules in Rectangular Pools 4.
Drawings to assist in evaluation cc w/encls:
See next page DOCUMENT NAME:
G:iGINNAiM95759.MTS IC To receive a copy of this document, indicate in the box:
"C" = Copy without attachment enclosure "E" = Co with attachment n losure "N" = No co OFFICE PH:PDI-1 E
LA:PDI-D HAHE GVissing/rsI SLitt AD c
)
DATE 09/
/97 09/
/97 09
/97 Official Record Copy
I, i,
I
5.
The deformation shape and magnitude of the deformations of the rack for the single-rack SSE analysis when the maximum displacement at the rack top corner occurs.
6.
The magnitude of the hydrodynamic pressure distribution along the height of the rack during the fluid and rack interaction for 3-D single and multi-rack analyses.
7.
Peak response results and stresses of the single and multi-rack SSE analyses.
8.
Impacts between a rack and concrete spent fuel pool wall and between racks.
9.
The quality assurance program to assure proper construction and installation.
The staff indicated that a more detailed request for additional information will be formally submitted following the meeting.
The licensee and its vendors representatives appeared to understand the concerns of the staff and indicated that they could respond to the formal questions within 30 days of receipt.
Sincerely, Guy S. Vissing, Sen' Project Manager Project Directora e I-1 Division of Reactor Projects I/II Office of Nuclear Reactor Regulation Docket No. 50-244
Enclosures:
1.
List of Attendees 2.
Results of the ANSYS code 3.
Technical paper on Dynamic Fluid-Structure Coupling of Rectangular Modules in Rectangular Pools 4.
Drawings to assist in evaluation cc w/encls:
See next page
Rochester Gas and Electric Corporation R.
E. Ginna Nuclear Power Plant CC:
Peter D. Drysdale, Senior Resident Inspector R.E.
Ginna Plant U.S. Nuclear Regulatory Commission 1503 Lake Road
- Ontario, NY 14519 Regional Administrator, Region I U.S. Nuclear Regulatory Commission 475 Allendale Road King of Prussia, PA 19406 Hr. F. William Valentino, President New York State
- Energy, Research, and Development Authority Corporate Plaza West 286 Washington Avenue Extension
- Albany, NY 12203-6399 Charlie Donaldson, Esq.
Assistant Attorney General New York Department of Law 120 Broadway New York, NY 10271 Nicholas S.
Reynolds Winston 5 Strawn 1400 L St.
N.W.
Washington, DC 20005-3502 Hs. Thelma Wideman
- Director, Wayne County Emergency Management Office Wayne County Emergency Operations Center 7336 Route 31
- Lyons, NY 14489 Ms. Mary Louise Heisenzahl Administrator, Monroe County Office of Emergency Preparedness 111 West Fall Road, Room 11 Rochester, NY 14620 Dr. Robert C. Hecredy Vice President, Nuclear Operations Rochester Gas and Electric Corporation 89 East Avenue Rochester, New York 14649 Mr. Paul Eddy New York State Department of Public Service 3 Empire State Plaza Tenth Floor
- Albany, NY 12223
Meetin Summar Distributi on E-Mail Enclosure 1 only S. Collins/F. Miraglia R.
Zimmerman (RPZ)
B. Boger A. Dromerick G. Vissing S. Little T. Martin (SLM3)
W. Beckner Y. Kim R.
Rothman G. Bagchi L. Doerflein, RI B.
McCabe (BCM)
~Hard Co
~Doc~ket Fi.le w/all enclosures PUBLIC - w/all enclosures PDI-1 R/F w/all enclosures OGC/w Encl.
1 ACRS/w Encl.
1
LIST OF ATTENDEES ROCHESTER GAS AND ELECTRIC CORPORATION R.
E.
GINNA'NUCLEAR POWER PLANT AUGUST 25, 1997 NAME Francis Marquet Mahendra K. Punater John R. Biddle Robert Borsum
'en Sucheski Jaime Ortiz George Wrobel Y. S.
Kim Robert Rothman Guy S. Vissing ORGANIZATION Framatome/ATEA Framatome Cogema Fuels Framatome Cogema Fuels Framatome Cogema Fuels RG&E RG&E RG&E NRC/NRR/EGCB NRC/NRR/EGCB NRC/NRR/PD I-1 Enclosure 1
1]
1, 1
APPENDIX V.M V24 Verification of Hydrodynamic Coupling Element {STIF38)
To verify the accuracy of the hydrodynamic coupling element (STIF38), two simple problems are studied.
Each problem consists of a the same two-mass system, as shown in the figure below, the first "without" hydrodynamic coupling and the second "with"hydrodynamic coupling taken into account.
Oi O>
The model consists oftwo massless rigid beams(Elements 1 &2), each fixed in a11 DOF at one end, with masses(m, & mQ lumped at the free ends.
The following properties are used:
A, = 5 in'
= 54.5 in', = 1600 lbs = 4.1408 lbs-sec'/in, E = 30E6 psi A> = 256 in~, I> = 54S9 in", m> = 100 lbs = 0.2588 lbs-sec /in, E = 30E6 psi The ANSYS results for the first case(without hydro. coupling) are shown in Table V.M.1. The frequencies of the first two modes are 2.4736 sec 'nd 98.9333 sec'.
This can be checked by hand computations as follows:
= 2.473 sec 1
2m mi 2'
~ 1408
- where, (3) (30E6) (S4
~ 6) g ppp ~~+/
(170) '
S&W Fuel Company Page V.M-1 51-1228073-01 Enclosure 2
Similarly, f~
98.933 sec
~
2' 2m 0.2588
- where, X
+
54
= 100 000 2bs/in (170)
'he basic hydrodynamic coupling equations are derived for an infinitely long cylinder concentrically contained within another infinitelylong cylinder, with a gap offiuidin between.
The necessary additional inputs to ANSYS are:
R, = radius of outer body = 4.942 in. (arbitrary)
Ri = radius of inner body = 4.7956.in. (arbitrary) h = length(height) of each body The ANSYS results for the first case(with hydro. coupling) are shown in Table U.M.I. The frequencies of the first two modes are 0.77391 sec-I and 8.1199 sec-I.
This can be checked by hand computations as follows(Ref. 10, Singh etal.):
f~
- 0 7739 sec
~
2m mi+M~
28 (4 1408+38.16)
- where, Hz [
]PSRg~
= [
] [9.35Z-5] Bc] [4.7956'] [170]
38.16 2bs-sec'n Similarly, B&WFoeI Company Page V.M-2 51-1228073-01
- 8.11885 sec
's
~+&1>>
2s (0.2588+38
~ 16)
Therefore, it has been shown by simple example that the results of the ANSYS models, using STIP 38 elements, agree with published equations.
SRYV Fuel Company Page V.M-3 51-1228073-0i
~
~
Table V.M.i ANSYS Results for Two Mass System Without Hydrodynamic Coupling NUMBER OF ELEMENT TYPES=
4 NUMBER OF REAL CONSTANT SETS =
4 MAXURUMMATERIALNUMRER= I NUMBER OF SPECIFIED MASTER D.O.F. =
TOTALNUMBER OF MASTER D.O.F.
2 2
~ ~~* CENTROID, MASS, AND MASS MOMEN'I'S OF INERTIA ~~*~~
CALCULATIONSASSUME ELEMENT MASS AT ELEMENT CENTROID TOTAL MASS =
4.3996 MOM. OF INERTIA MOM. OF INERTIA CEN'IROID ABOUT ORIGIN ABOUT CENTROID XC =
.00000E+00 YC =
170.00 ZC =
.00000E+00 IXX=
IYY=
IZZ =
IXY=
IYZ =
IZX =
.1271E+06
.0000E+00
.1271E+06
.0000E+00
.0000E+00
.0000E+00 IXX=
IYY =
IZZ =
IXY =
IYZ =
IZX =
.1904E-11
.0000E+00
.1904E-.11
.0000E+00
.0000E+00
.0000E+00
~~~ MASS SURGERY BY M3":MENTTYPE ~~*
TYPE MASS 3
.258800 4
4.14080 RANGE OF ELEMENT STIFFNESS IN GLOBALCOORDINATES MAXMUM=.33334IETIU AT ELEMENT I.
MINIMUM=.385412E+08 AT ELEMENT 2.
BSc% Fuel Company Page U.MQ 51-1228073-01.)
~~~~~ EIGENVALUE(NATURALFREQUENCY) SOLUTION ~*
- MODE FREQUENCY (CYCLES/TIME) 1 2.47355857 2
98.9332817
~~~~~ REDUCED MASS DISTRIBUTION ~*~~~
ROW NODE DIR VALUE 1
UX
.25880 2
2 UX 4.)408 MASS(X,Y,Z) =
4.400
.0000E+00
.0000E+00
~*~**PARTICIPATIONFACTOR CAI.CULATION~~~~~ X DIRECTlON CUMULATIVE-MODE FREQ.
PER.
PART.FACT.
RATIO EFF. MASS MASS FRACT.
2.47356 98.9333
.40428 2.0349 1.00000
.10108E-01
.50872
.250000 4.14080
.258800
.941176 1.00000 MODE SUM OF EFFECTIVE MASSES=
4.39960
~~**~PARTICIPATIONFACTOR CALCULATION~*~** Y DIRECTION CUMULATIVE FREQ.
PERIOD PART.FACT.
RATIO EFF. MASS MASS FRACT.
2.47356
.40428
.00000E+00 98.9333
.10108E-01
.00000E+00
.000000
.000000
.00000
.00000
.00000
.00000
~~~~* PARTICIPATIONFACTOR CALCULATION~~~~~ Z DIRECTION CUMULATIVE MODE FREQ.
PERIOD PART.FACT.
RATIO EFF. MASS MASS FRACT.
1 2.47356 A0428
.00000E+00
.000000
.000000 2
98.9333
.10108E-01
.00000E+00
.000000
.000000
.000000
.000000 AA%LW
. )
B&WFuel Company Page V.M-5 51-1228073-01
Table V.M,2 ANSYS Results for Two Mass System With Hydrodynamic Coupling 2
2 NUMBER OF ELEMENT TYPES=
6 NUMBER OF REAL CONSTANT SETS=
6 NUMBER OF NLEMENFB =
6 MAXMIEA MAXNIIMMAFERIALNUMBER=
6 NUMBER OF SPECIFIED MASTER D.O.F. =
TOTALNUMBER OF MASTER D.O.F.
NODE NUMBER USED =
5
~~~*~ CENTROID, MASS, AND MASS MOMENTS OF INERTIA ~~~~~
CALCULATIONSASSUME ELEMENT MASS AT ELEMENT CEN'IROID TOTALMASS =
4.5418 CENTROID MOM. OF INERTIA ABOUT ORIGIN MOM. OF INERTIA ABOUT CENTROID XC =
YC =
ZC =
.00000E+00 IXX=
.1313E+06 170.00 IYY~
.0000E+00
.00000E+00 IZZ =
.1313E+06 IXY=
.0000E+00 IYZ =
.0000E+00 IZX =
.0000E+00 IXX=
.1947E-10 IYY=
.0000E+00 IZZ =
.1947F 10 IXY=
.0000E+00 IYZ=,.0000E+00 IZX =
.0000E+00 ONLYTHE FIRST REAL CONSTANT MASS TERM IS USED FOR THE STIF21 ELEMENrS.
ONLYTHE FIRST REAL CONSTANT MASS TERM IS USED FOR THE STIF21 EUMENTS.
~*~ MASS
SUMMARY
BY ELEMENT TYPE ~*~
TYPE 3
4 5
6 MASS
.258800 4.14080
.711113E-01
.711113E-01 RANGE OF ELEMENT STIFFNESS IN GLOBALCOORDINATES B&WShel Company Page V.M-6
~
51-1228073-01
hGOCMM=.385341E+10 AT ELEMENT 1.
MINMUM=.385412E+08 AT ELEMENT 2.
¹¹¹¹¹ EIGENVALUE(NATURALFREQUENCY) SOLUTION ¹¹*¹¹ MODE FREQUENCY (CYCLESf%ME) 1
.773907461 2
8.11991626
¹¹***REDUCED MASS DISTRIBUTION¹¹¹*¹ RO% NODE DIR VALUE 1
1 UX 38.419 2
2 UX 42.301 MASS(X,Y,Z) =
80.72
.0000E+00
.0000E+00
¹¹¹*¹ PARTICIPATIONFACTOR CALCULATION¹¹¹*¹ X DIRECTION CTEQJLATIVE MODE FREQ.
PERIOD PART.FACT.
RATIO EFF. MASS MASS FRACT.
1
.773907 1.2921 6.5039 1.000000 42.3010
.524046 2
8.11992
.12315 6.1983
.953011 38.4190 1.00000 SUM OF EFFECI'IVE MASSES=
80.7200
- ¹*¹PARTICIPATIONFACTOR CALCULATION¹¹*¹* Y DIRECTION CUMXJLATIVE MODE FREQ.
PERIOD PART.FAC.
RATIO EFF. MASS MASS FRAG.
1
.773907 1.2921
.00000E+00
.000000
.000000
.00000 2
8.11992
. 12315
.00000E+00
.000000
.000000
.00000
¹¹¹¹¹ PARTICIPATIONFACTOR CALCULATION¹¹*¹¹ Z DIRECTION CUMULATIVE MODE FREQ.
PERIOD PART.FACT.
RATIO EFF. MASS MASS FRACT.
1
.773907 1.2921
.00000E+00
.000000
.000000
.000000 2
8.11992
.12315
.00000E+00
.000000
.000000
.000000 BEcYV Brel Company Page V.M-7 S1-1228073-01
'E I
5, l
C 0
)
DYNAMICFLUID STRUCTURE COUPLING OF RECTANGULAR MODULES IN RECTANGULAR POOLS R. J. Scavuaao Professor and Head, Oepartment of Mechanical Engineering The University of Akron Akron, Ohio
~
W. F. Stokey Associate Professor of Mechanical Engineering Carnegie Mellon University Pittsburgh, Pennsylvania E. F. Radke Senior Stress Engineer 8abcock and Wilcox Company 8a rberton, Ohio Abstract This paper presents a normal mode analysis of a structure in a pool sub-ject to seismic loading.
Supporting experimental data is presented.
Fluid coupling between the pool walls and structure is taken into account.
The struc-ture is assumed to be cantilevered from its base.
Equations to calculate struc-ture loads and hydrodynamic forces on the pool are developed.
These equations show that the modal participation factors are modified when compared to the usual form.
In addition, the virtual mass for a rectangular module in a rec-tangular pool with plane fluid motion is developed.
Resulting equations for the virtual mass and the calculated dynamic displacement are experimentally verified.
Introduction Often structures are submerged in a fluid and subject to transient founda-tion motion.
Excitation of the structure occurs through the foundation and by dynamic coupling through the fluid.
In this paper a normal mode solution to this problem is developed.
The resulting equations are compared to some experi-mental results.
In addition, the virtual mass for a rectangular module in a rectangular pool is developed.
The solution presented in this paper is an extension of the work of Fritz and previous work of the authors (1,2).'he fluid is assumed to be incompres-sible and viscous effects are neglected.
Fluid coupling between the pool walls and structure is caused entirely from fluid inertia.
Coupling is based on element to wall motion; interelement coupling is neglected.
Normal Node Theory yi Hi li Hi Equations of Notion Based on the assumptions
- above, the coupling body and an element of a rigid foundation through (1,2) between an element of a rigid a fluid can be written as
~I
~t Yi Z
+
0 Z
Enclosure 3
where (Fig., 1.0)
Ff.
= Force aching on the pool wall by the i mass
, F
= Force acting on the i mass yi HHi Hydrodynamic mass associated with m,.
M Hass of the fluid displaced by m.
li 1
H2,.
Hass of fluid based on the volume enclosed by the pool wall which is coupled to mi Yi(t)a Motion of mi relative to the pool wall foundation Z(t)
Absolute foundation motion
.Therefore the fluid force on each mass is F1
'Hi"i 'li'2) or for all masses (F )
-[MM](Y)+ [M13(Z)
(3)
If [K] is the stiffness matrix and [m] the mass matrix of the structure, the equat;ion of motion is I ml L(Y} + {Z}]
[K3(Y}
(Fy'}
(4)
Thus, substituting for (F )
((m] + [MH))(Y)+ [K3(Y)
-((m]-[Hl))(Z}
(5)
Modal Displacement The eigenvalues.
ma~, and eigenvectors P ) of the homogeneous equation can be determined in the usual manner.
Let (Y}
E{V ) q (t) a l2g80&S Using the orthoganality principle, equation (5) becomes
~0
~0 where (Y,) ((m]-[H133(l)
P.
a a
(7 )T((m+ M(Y Fluid coupling between the structure wall reduces both the natural frequencies and the modal participation factor when compared to the uncoupled structure.
The magnitude of the change depends upon [Ml] and
[MH].
These effective masses depend on the mass of the displaced fluid and clearances in the coupled system.
If the initial conditions are zero (qa(0) qa(0)
- 0) (3 )
P q (t) a - J z(v) sin (u) (t-~))dv (g)
Pa"a a max where V
is the spectrum velocity at the circular frequency, m
( 1 0)
Structure Modal Forces The forces acting on the structure can be written as (12)
Mall Modal Forces
{F ) = IK] {Y} > [K] Z {Y ) q (11)
Substitution into equation (ll) yields the following
( y)
II 3 IHH]3 If the mass matrix is diagonal, the force acting on each mass, mi, in each mode can be expressed in terms of the spectrum velocity, Va, as follows yfa i
Hi a"ai "a"a (13)
Forces acting on the pool wall can be written as (Ff)
IIMl ] + IMH))(y}- {'23(1) Z Since (Y} - Z (Y )q - Z (Y )I-p Z - m'q 3
(14) and it can be shown that (3)
E (7,} P, - (1}
a the forces acting on the wall can be written as (16)
(Ff) = - IIM13+ IMH))E Ya} "a"a IIM23+ IH13+ IHH3]{l}Z The first term is associated with the dynamic response of the structure.
Each modal displacement (Ya)qa fs multiplied by ~a.
The second term is usually domfnated by the total mass of fluid in a pool, IH2] accelerated by the founda-tion motion, Z(t).
Exper imental Ver ffication In the Appendix of this paper, the hydrodynamic mass of an infinite rec-tangular module in an infinite rectangular pool is developed.
This result was verified experimentally by measuring changes in the resonant response frequency.
In addition, the basic coupling theory between the pool and the structure is verfffed by comparing the calculated displacement amplitude to measured values for a sinusoidal foundation input. (g)
Model Oescrfption In this experimental study a 4" x 4" x 3/16" (101.6 x 101.6 x 4.76 ma) structural tube eight fnclies. long (203.2 mn) was used to model a rigid box.
(Figure 2.0)
This tube was supported by two high strength steel cantilever beams 3/32" x 1" x 4" (2.38 ma x 25.4 eo x 101.6 nrn) which served as springs and were attached to a rigid foundation.
Because of the parallelogram effect, the~e was no significant rotation of the tube during horizontal motion.
Seals at the top and bottom of the tube restricted vertical fluid flow.
Originally the fit of the teflon seals was tight, however, in order to eliminate friction (and thus Coulomb damping) a small clearance was provided.
As a result, some ver tical fluid motion was observed.
However, the basic motion of the fluid
~
~
~
~
~
~2-1228ogg during harmonic excitation was in the horizontal plane.
A fluid filled plexiglass tank or container surrounded the rectangular tube with a gap of 0.5 inches (12.7 mm) on all sides.
The foundation of the tank was fabricated from 1/4" 6.35 mn) steel plate.
This plate was reinforced with two steel angles.
The entire tank assembly was supported by cantilever beams from a rigid foundation.
An electrodynamic shaker was used to input a sinu-soidal motion to the tank foundation plate.
The effect of fluid coupling between the tank walls and rigid box was studied by measuring the response of the system with and without water in the tank.
The center of the rigid box was dry and not filled with water for all measurements.
During preliminary testing of the system, it was found that addi-tional weight had to be added to the box or tube in order to obtain relative
- motion between the tank and tube.
Without this additional weight the tube moved with the motion of the tank walls.
The total weight of the tube, additional weight, seals, bolts, etc.
was 15.0 lbs (6.8 kg).
Hotions of both the pool foundation and tube were measured with piezoelec-tric accelerometers.
Output was recorded on a storage oscilloscope.
By taking a ratio of the response to the input, a measure of displacement was obtained at each input frequency.
Hydrodynamic Hass Effects The amplitude ratio in air is presented on Figure 3.0.
The natural fre-quency of the system was found to be 15.2 Hz.
The damping ratio of the system was found to 'be approximately 5 percent based on the amplitude at resonance.
This frequency was used to determine the effective stiffness of the cantilever beams:
K 4m~f~ w/g where f ~ 15.2 Hz w "- 15.0 lbs (6.8 Kg) g 386 in./sec'980 nm/s~)
K ~ 354 lbs/in.
(62 N/mo)
(18}
After filling the tank with water, the response shown in Figure 4.0 was measured.
The experimental natural frequency was found to be 9.7 Hz. If the hydrodynamic mass is based on the box dimensions (rather than the pool center-line) good agreement between theory and experiment is obtained.
For this case 16 yhb (19) where h ~ 8 in. (203.2 am),
b ~ 2 in. (50.8 am),
G
- 1/2 in. (12.7 ttm),
A ~ 62.4 lb/ft~ (1 Kg/m')
Thus WH 24.65 lbs (11-1 Kg)
The calculated natural frequency is determined as follows (20) where K ~ 354 lb/in. (62 N/am), g ~ 386 in./sec~
(980 mm/s'),
W ~ 15.0 lbs (6.8 Kg), WH ~ 24.65 lbs (11.1 Kg)
- Thus, f ~ 93Hz This value compares well with the experimental value.
Because some fluid by-passes the seals and moves vertically, the measured natural frequency is ex-pected to be greater than calculated values.
Agreement between experiment and theory is felt to be very good.
Response
Curve Shape Because of the relatively small magnitude at resonance of the ratio of amplitudes, it was originally assumed that there was very high damping of the structure caused by the fluid motion.
The amplitude ratio was observed to approach one as the forcing frequency increase could not be explained.
By ap-plying the theory developed
- above, both of these phenomena can be explained with a relatively low damping ratio.
For a one mass system, the equation of motion of the structure can be writ-ten as
~0 I
(m + M) x + cx + kx ~ (H+ Ml } z + cz + kz (21) where x and z are both absolute motion.
By dividing by (m + H). equation (21}
becomes
~tx+ 2u>
Zx + uF x =
z +
2a>
Z z + uP z n
n m +Mn n
(22)
If x Xe~"
and z Ze~
Thus "H'1 g
hl
+ 2'l g $
+~>
n m
+
n Ce 3
Z (bl m ) + 2') 4lPQ n
n Therefore the amplitude ratio is (23) where x/.
D-m
+
(
)3 +(2
<)'+
Hl "n
"n 0 - (
") 3
+ (2 "
~)'n "n
(24)
(26)
Note as becomes large, equation (24) converges to the following mass ratio x
0.793 Z~
m
+H (26)
For the uncoupled
- case, the ratio goes to zero with increasing frequency (Fig.
3.0).
Results of equation (24) are plotted on Figure 5.0. It should be noted that the damping ratio is approximately 6.2 percent for a curve with the same
~
~
~
~
~
~
~ ~
magnification at resonance.
Furthermore, the graph has the same basic shape as, the experimental curve and the ratio increases with frequency.
This value of the damping ratio is only 1 percent greater than the ratio found for the vibra-tion of the structure in air.
This value is consistent with the theoretical work of Au-Yang and Skinner (5,6).
i Conclusions Normal mode equations are developed which may be used to calculate both the forces on the submerged structure and on the pool walls.
The usual spectrum inputs may be applied.
However, the modal participation factor of the structure is modified, The results of the experimental investigation verify the calculated hydro-dynamic mass for a square rigid structure in a square pool.
Furthermore, by comparing the measured and calculated response curves of the fluid coupled sys-tem, the basic coupling theory developed in the paper is verified.
The damping ratio of the coupled system was found to increase by only a small amount
("-1 percent).
However, this damping ratio of the coupled system is based on the sum of the structure mass and hydrodynamic mass.
As a result, peak damping forces acting on the coupled system for a given amplitude were shown to increase by 25 percent above values acting on the uncoupled system.
Acknowledgements A portion of the work presented in this paper was done under the sponsor-ship of Machter Associates Inc., Pittsburgh, Pennsylvania.
The authors express their appreciation for permission to publish this work.
Mr. Steven Meissberg, of Rachter Associates, originally developed the equation of the hydrodynamic mass for a square module in a square pool using a different procedure, References 1
- Fritz, R. J,.
"The Effect of Liquid on the Dynamic Notions of Imersed Solids," Transactions ASNE Journal of Engineering for Industry, February 1972.
2
- Stokey, M. F., and R. J.
- Scavuzzo, "Normal Node Solution of Fluid-Coupled Concentric Cylindrical Vessels,"
ASHE Publication ?7-PVP-37, 1977.
3 Cunniff, P.
E. and G. J. O'ara, "Normal Node Theory for Three Dimen-sions,"
NFL Report 6170. Naval Research Labor'atory, Rashington.
- 0. C.. 1965.
4
- Radke, Edward F., "Experimental Study of Immersed Rectangular Solids in Rectangular Cavities,"
Prospect for Master of Science
- Degree, The University of Akron, Akron, Ohio, 1978.
5 Au-Yang, N. K. and D. A. Skinner, "Effect of Hydroelastic Coupling on the Response of a Nuclear Reactor,to Ground Acceleration," Paper K5/5, 4th International Conference on Structural Mechanics in Reactor Technology, 1977.
6 Au-Yang, N. K., "Response of Fluid Elastically Coupled Coaxial Cylin-drical Shells to External Flow." J.
Fluids Engr., Vol. 99, p 319, 1977.
Appendix Hydrodynamic Mass It is the objective of this appendix to present a derivation of the hydro-dynamic mass for a square rigid module in a square pool.
It is assumed that the gap surrounding the module is narrow so that approximations used for fluid velocities are justified.
Furthermore, it is assumed that these gaps are uni-form and the geometry is symetric so that the flow velocity at the center of gap (y = 0) is zero (Fig. 6.0).
In addition, flow in the axial direction is neglected.
Only motion in the xy plane is considered for a structure of height, h.
Fluid Velocities The fluid velocity in the y direction of the top gap is
4 "1
The magnitude is identical in the bottom gap.
The velocity in the gaps parallel to the x-axis is X2 Xl Y
(
)
b X
W2 Kinetic Energy (Al )
(A2)
(A3)
Fluid Force Hatrix Since the fluid forces can be determined from the following (1):
The kinetic energy of the fluid in the channels of height, h, caused by relative motion is b
1 2
X2-Xl 2 T = 4f pY w h dy + 2p
(
)
b chw 2
y l
w2
'~
2 F. - (
)
d T
i dt i
the forces from relative motion become g/
V~-
(A4) xl where x2 "2
~ 4 b'h + ~4b~ch "H
3 wl w2 If b c and wl w2 w,
16 b~h
.h-6 h~
--"~ Zg (AS)
(A6)
(AT)
Total Hydrodynamic Nass Matrix 8y considering absolute motions of the system, two additional hydrodynamic masses are developed (1):
and 4p (b - w/2)(c - w/2)h M
4p (b + w/2)(c +'w/2)h (As)
(Ag)
Thus. including these effects Fx2
~w Xl
~w X2 (AlO)
~ f lear l4II hallos Ffg.
1 NTHEMATICAL HODEL MITH FLUID COUPLING TO THE MALL gas rugZ
~ N4E'AIGLAS~
E'R r ~erz~s ffOrCDN
~~rex c uvre OVCRCAPJhiS rCFLo)V sE'ALM SHAPE'R s TE'E L supposer S~RIrVD5 4
l 1
4 4
4 J
a C
0 d
c&lcÃEPF SLCC'h Ffg.
2 EXPERIMENTAL APPARATUS
2-122S095 10 10
'IS
- f. (IIz) 20 25 35
- Fig, 3
MEASURED AMPLITUDE GF THE STRUCTURE IN AIR
'0 eo r
Eaorrlaenaal Rrarlt
theoretical Curve for C >.062 Vafn9 Theoretical f 3
'IO IS 20 f, (Nz) 25 Fjg.
4 MEASURED AND CALCULATED ABSOLUTE AMPLITUDE RATIO OF THE STRUCTURE IN MATER
~~28095 10 C ~ 0 C ~. 06t C e
~ t1 Ffg. 5 CALCULATED ABSOLUTE AMPLITUDE RATIO OF THE STRUCTURE IN WATER WITH DIFFERENT DAMPING RATIOS 2b Xt 2c M)
Fig 6
RECTANGULAR STRUCTURE IN RECTANGULAR POOL W(
- .Dj
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