ATOMIC ENERGY CjSSft L'ÉNERGIE ATOMIQUE OF CANADA … · Dynamique d'assemblages combustibles dans...
Transcript of ATOMIC ENERGY CjSSft L'ÉNERGIE ATOMIQUE OF CANADA … · Dynamique d'assemblages combustibles dans...
AECL-5976 UKAEA ND-R-127(S)
ATOMIC ENERGY C j S S f t L'ÉNERGIE ATOMIQUEOF CANADA LIMITED V j f i ^ V DU CANADA LIMITÉE
DYNAMICS OF NUCLEAR FUEL ASSEMBLIES IN VERTICAL
FLOW CHANNELS:COMPUTER MODELLING AND ASSOCIATED STUDIES
Dynamique d'assemblages combustibles dans des canaux àécoulement vertical :
modélisation sur ordinateur et études connexes
V.A. MASON, M.J. PETTIGREW, G. LELLI, L. KATES and E. REIMER
Chalk River Nuclear Laboratories Laboratoires nucléaires de Chalk River
Chalk River, Ontario
October 1978 octobre
ATOMIC ENERGY OF CANADA LIMITED
Chalk Rlve.n Uu.clo.aA Labonato/Uu
DYNAMICS OF NUCLEAR FUEL ASSEMBLIES IN VERTICAL FLOW CHANNELS:
COMPUTER MODELLING AND ASSOCIATED STUDIES*
V.f\. M a s o n S M . J . Vettiavp.to, G. Lelli**,
L. KatP.s, K. J\r:ir»er
Engineering Research BranchChalk River, Ontario KOJ 1J0
October 19 78
* On attachment from the Sppin<jfi<>lls Nuclear' Poucr1 Development Labora-tories > U.K.A.E.A., SaUnck, Preston, England.
** ;'>: attachment from the. Comitatv "iasi-onalc Enevgia Nueleave, CentvoStudi Nualeav',, Casaoaia, Fore, Italy, in 1978.
AECL-5976
UKAEA ND-R-1?7(S)"•/'••.;: rr>n<->r>- •.'.••; /'.mnir-f joini.l. < bij Ainnict Enrvnii of Canada Limit adniid .'>;./ /)'/•-• H>;!j,:d Klnqd"rn At<vnic Kurrna Auihrri tn.
Dynamique d'assemblages combustibles dans des canaux à écouleirent vertical:
modélisation sur ordinateur et études connexest
par
V.A. Mason*, M.J. Pettigrew, G. Lelli**,
L. Kates et E. Reimer
Résume
On décrit danc ce rapport un modèle mathématique d'ordinateur conçu pourprédire le comportement dynamique d'assemblages combustibles assujétis à unécoulement axial.
Les méthodes numériques utilisées pour établir et résoudre les équationsmatricielles du mouvement dans le modèle font l'objet de commentaires et ondonne un aperçu de la méthode employée pour interpréter les données destabilité des assemblages combustibles. On décrit en détail les mathématiquesdéveloppées pour les calculs à réponse forcée.
Certains paramètres de la modélisation structurelle et hydrodynamique ontdû être déterminés expérimentalement. On identifie ces paramètres et ondécrit brièvement les méthodes employées pour les évaluer.
On donne à la fin du rapport des exemples d'applications typiques du modèledynamique.
L'Energie Atomique du Canada, LimitéeLaboratoires nucléaires de Chalk River
Chalk River, Ontario KOJ U O
t Ce rapport est publié conjointement par l'Energie Atomique du Canada,Limitée et par the United Kingdom Atomic Energy Authority.
* Détaché des Springfields Nuclear Power Development Laboratories, U.K.A.E.A.,Salwick, Preston, England.
** Détaché en 1975 de Ccmitato Nazionale Energia Nucleare, Centro StudiNucleari, Casaccia, Rome, Italy.
AECL-5976
UKAEA ND-R-127(S)
Octobre 1978
ATOMIC ENERGY OF CANADA LIMITED
Chalk RlveA Nu.cle.an.
DYNAMICS OF NUCLEAR FUEL ASSEMBLIES IN VERTICAL FLOW CHANNELS:
COMPUTER MODELLING AND ASSOCIATED STUDIESf
by
V.A. Mason*, M.J. Pettigrew, G. Lelli* *,
L. Kates, E. Reimer
ABSTRACT
A computer model, designed to predict the dynamic behaviour of nuclearfuel assemblies in axial flow, is described in this report.
The numerical methcds used to construct and solve the matrix equationsof motion in the model are discussed together with an outline of themethod used to interpret the fuel assembly stability data. The mathe-matics developed for forced response calculations are described indetail.
Certain structural and hydrodynamic modelling parameters must be deter-mined by experiment. These parameters are identified and the methodsused for their evaluation are briefly described.
Examples of typical applications of the dynamic model are presented towardsthe end of the report.
Engineering Research BranchChalk River, Ontario, KOJ 1J0
* On attachment from the Springfields Nuclear Power Development Laboratories,U.K.A.E.A., Salwiok, Preston, England.
** On attachment from the Comitato Nasionale Energia Nualeare, Centro StudiNualeari, Casaaaia, Rome, Italy, in 1975.
+ This report is issued jointly by Atomic Energy of Canada Limitedand by the United Kingdom Atomic Energy Authority.
AECL-5976
UKAEA ND-R-127(S)
October 1978
CONTENTS
Page
1. INTRODUCTION 1
2. THE DYNAMIC MODEL 4
2.1 Structural Kinetic Energy 52.1.1 Matrix Formulation 72.1.2 Computation 9
2.2 Kinetic Energy of Fluid 102.2.1 Matrix Formulation 122.2.2 Computation 14
2.3 Gravitational Potential Energy of the Structure 152.3.1 Matrix Formulation 162.3.2 Computation 16
2.4 Strain Potential Energy 172.4.1 Matrix Formulation 172.4.2 Computation 18
2.5 Structural Damping Forces 192.5.1 Matrix Formulation 192.5.2 Computation 23
2.6 Hydrodynamic Forces 242.6.1 Longitudinal Friction Forces 242.6.2 Normal Viscous Forces 252.6.3 Longitudinal Pressure Drop Forces 262.6.4 Lateral Pressure Forces 272.6.5 Base Drag Force 272.6.6 Non-conservative End Force 28
2.7 Generalized Moments 292.7.1 Matrix Formulation 312.7.2 Computation 37
2.8 Equations of Motion 392.8.1 Matrix Formulation 392.8.2 Computation 41
2.9 Solution of the Equations of Motion 422.9.1 Matrix Formulation 422.9.2 Computation 44
CONTENTS(Cont'd)
Page
3. INTERPRETATION OF STABILITY RESULTS 45
3.1 Eigenvalues 45
3.2 Eigenvectors 46
3.3 Modal Pairs 47
4. CONSTRAINED FUEL ASSEMBLY 50
4.1 Coordinate Reduction Matrix 51
4.2 Application of Matrix R 544.2.1 Theory and Matrix Formulation 544.2.2 Computation 56
4.3 Stiffness and Extra Damping at the End Constraints 574.3.1 Theory and Matrix Construction 574.3.2 Computation 61
5. TRANSIENT RESPONSE TO HARMONIC FORCES 62
5.1 Transient Response Theory 635.1.1 Uncouple the Equations of Motion 635.1.2 Solve the Uncoupled Equations 675.1.3 Generalized Harmonic Forces 71
5.2 Computation 73
6. STEADY STATE RESPONSE TO HARMONIC FORCES 74
6.1 Theory and Matrix Formulation 74
6.2 Computation 76
7. APPLIED FORCES IN A CONSTRAINED SYSTEM 77
7.1 Theory and Matrix Formulation 77
7.2 Computation 78
8. TRANSFER FUNCTIONS 79
8.1 Theory and Matrix Formulation 81
8.2 Computation 82
9. RANDOM EXCITATION 84
9.1 Construction of the Square Receptance Matrix 85
9.1.1 Theory and Matrix Formulation 859.1.2 Computation 86
CONTENTS(Cont'd)
Page
9.2 The Response to Discrete Force Spectra 869.2.1 Complex Response Spectrum 879.2.2 Power Spectrum of Response 879.2.3 Mean Square and Root Mean Square Values of Response 889.2.4 Computation 89
9.3 Response to Forces Expressed in Averaged Power SpectralDensity Format 909.3.1 Power Spectral Density of Response 909.3.2 Mean Square and Root Mean Square Values of Response 969.3.3 Computation 97
9.4 Response to Uncorrelated Forces 989.4.1 Theory and Matrix Formulation 989.4.2 Computation 100
10. RESOLVED FORCE CALCULATIONS 101
10.1 Calculation of Discrete Force Spectra 10210.1.1 Theory and Matrix Formulation 10210.1.2 Computation 104
10.2 Resolved Forces Calculated from Power Spectral Density ofResponse Data 10510.2.1 Theory and Matrix Formulation 10510.2.2 Computation 106
10.3 The Inverted Receptance Matrix 10710.3.1 Theory and Matrix Formulation 10710.3.2 Computation 109
11. LAMPS MATRIX PROCESSOR LANGUAGE 110
12. MEASUREMENT OF THE DYNAMIC MODELLING PARAMETERS 114
12.1 Structural Parameters 114
12.2 Fluid Parameters 116
12.2.1 Hydrodynamic Mass per Unit Length M and Viscous DragCoefficient Cp 116
12.2.2 Friction Coefficient 12312.2.3 Base Drag Coefficient, CB 12412.2.4 Free End Factor f 125
CONTENTS(Cont'd)
Page
13. DYNAMIC MODELLING EXAMPLES 127
13.1 Natural Frequencies and Mode Shapes 127
13.2 Travelling Waves I 2 8
13.3 The Effect of Flow Rate on Fuel Assembly Stability 129
13.4 Changes in the Hydrodynamic Parameters 130
13.5 Natural Frequencies and Mode Shapes of Uniform Beams 131
13.6 Transient Forced Response of a Uniform Cantilever 132
13.7 Steady Harmonic Response 133
13.8 Transfer Functions 135
13.9 Response to Random Forces 135
13.10 Resolved Force Calculations 136
14. CONCLUSION 138
15. REFERENCES 139
FIGURES 1 - 49 141-185
APPENDICES
Appendix I - DYNMOD Listing 186
Appendix II - Matrix Identities and Operations 212
Appendix III - Structural Damping: An Alternative Implementation..215
Appendix IV - Interbundle Parameter Gradients 22^
Appendix V - Lamps Functions and Subroutines used in DYNMOD .. 226
TABLE 1 233TABLE 2 227
-1- AECL-5976
1. INTRODUCTION
Severe mechanical damage can result from the excessive vibration
of nuclear fuel assemblies in vertical flow channels. Fretting
between fuel assembly and channel, pressure seal failures and
the fatigue failure of component parts are examples of the problems
caused by flow-induced vibration.
Dangerous fuel assembly vibrations are often detected during
in-reactor and environmental loop testing experiments. A more
quiescent fuel and channel system is eventually developed after
a lengthy series of ad hoc modifications and tes ts . If the dynamic
characteristics of a prototype fuel assembly could be predicted
at the design stage, considerable development effort and expense
could be avoided.
A computer model of the dynamics of nuclear fuel assemblies in
axial flow has been developed at A.E.C.L., Chalk River. The dynamic
model uses matrix operator calculus and is based on the analytical
expressions formulated and published by M.P. Païdoussis (réf. 1).
It is intended that the model be used as a design tool for the
development of dynamically well-behaved nuclear fuel assemblies.
The computer program was written originally for the examination
of CANDU-BLW1reactor fuel string vibrations. However, the work
is also directly relevant to the vertical booster fuel rods of
the CANDU-PHW2 reac to r and to the fuel assemblies of both the
Bri ish SGHWR3and Italian Cirene reactor. The fuel assembly
dynanics of various other nuclear reactor systems, such as the
fast breeder reactor, can be investigated using a slightly
modified program. In addition, the model can be used in the
non-nuclear field to predict the vibration characteristics
and dynamic s tabi l i ty of pipework, heat exchanger components,
and self-oropelied cylindrical bodies moving in air or water.
1 CA-W'-BLW - CANada Deuterium Uranium - Boiling Light Water2 CANDU-PHW - CANada Deutarium Uranium - Fpessurized Heavy Water3 SGHWR - Steam Generating Heavy Water Reaotor.
-2-
The dynamic modelling program (DYNMOD) calculates the natural
frequencies and mode shapes of reactor fuel assemblies in
axial flow. This information describes the free vibration
characteristics and dynamic stability of the fuel.
DYNMOD predicts the transient response of fuel assemblies to
harmonic forces and the steady state response to these forces.
It also calculates the mobilities and receptances of the fuel
string. These transfer functions contain amplitude and phase
information and are evaluated over a range of frequencies.
The response of a fuel assembly to random forces and complicated
forcing functions can be evaluated by DYNMOD. The applied
forces are expressed as discrete spectra or averaged power spec-
tral densities in correlated or uncorrelated form. In all cases,
structural response can be calculated with or without axial flow.
Where external forces are involved, the response to any number
of forces of any amplitude and phase (where applicable) can be
evaluated.
The program will deduce the nature of applied forces acting on
a fuel assembly. This requires a detailed knowledge of the
response and transfer function characteristics of the fuel string.
The transfer functions can be calculated by DYNMOD or input from
experimental data.
This report describes the mathematical formulation of the dynamic
model. In addition the nature and interpretation of fuel stability
data are discussed. A short description of the LAMPS matrix
processor language, which was used to handle the numerous matrix
operations in the model, is included in the report. After des-
cribing the experimental methods used to define the hydrodynamic
modelling parameters, a number of examples of the use of DYNMOD
- 3 -
are presented. (Some examples discuss the response and stabil i ty
characteristics of CANDUBLW and UKAEA-SGHW reactor fuel
assemblies. The results presented in these cases are purely
hypothetical and are not relevant to current or proposed fuel
designs.)
Algebraic variables and matrix operators used in the test
are listed and defined in tables 1 and 2. A l is t ing of DYNMOD
is presented in Appendix I.
-4-
2. THE DYNAMIC MODEL
It is assumed that the fuel assembly to be modelled can be
represented by an articulated structure consisting of a string
of interconnected fuel bundles as shown in Figure 1. Each bundle
is described in terms of its mass, geometric dimensions, inherent
structural damping and hydrodynamic mass. The structural
stiffnesses of the assembly are introduced into the dynamic model
as interbundle bending stiffness (K , central support tube
stiffness in BLW reactor fuel), bundle endplate stiffness (KENn)>
and bundle parallelogramming or shear stiffness (KpAR)•
The equations of motion of the idealized fuel assembly are
generated by the substitution of energy and force terms into
the non-conservative Lagrangian equation
(Ml)
Here T and V are the kinetic and potential energies of the
vibrating fuel/fluid system at time t. The column vector Q
describes the generalized forces acting on the fuel assembly
and q describes the generalized coordinates of the system.
The matrix equation (Ml) is solved in terms of two sets of
generalized coordinates 0 and iJJ. >j) represents the angular
deflections of the bundles from the vertical, and ijJ describes
the angular deflections of the bundle end faces from the
horizontal. For an assembly of N bundles, <i> and ijJ are of order
(N x 1), and the vector q,(2N x l),is defined as
— (dt
( 3 T \I
i - ;
^ 3 q /
9T_ -—
9q+
oV—9q
The energy terras introduced into Lagrange's equation are the
gravitational and strain potential energies (V and V )
of the fuel assembly, the kinetic energy of the structure (T
and the lateral kinetic energy of the fluid entrained by the
-5-
fuel string (T ). The generalized hydrodynamic forces (Q )
are derived from the pressure drop forces, the normal and
longitudinal friction forces, and the lateral drag forces along
the fuel assembly. Base drag at the end of the fuel assembly
and a non-conservative inviscid force, attributed to the loss
in lateral momentum flux at the free end, are also included in_ /IT \
Q . Structural damping is incorporated in the model as thegeneralized force vector Q .
Because T = T ^ + T^f* and Q = Q*H* + Q^\ equation (Ml) can
be written as
d /3T(s\ d /3T(f)\ 3T(s) 3T(f) 3V( ) ( Q(H) + Q(D) (M2)
dt \ 3q / dt \ 3q / 9q 3q 3q
In the following sections, each of the seven terms in (M2) is
expressed as a function of q, q, and q. Addition of the resulting
coefficient matrices gives an equation of the form
A • (|) + B • $ + C • (J) = 0 (M3)
Solution of this equation gives the natural frequencies and
mode shapes of the fuel assembly in axial flow.
2. 1 Structural Kinetic Energy
With reference to Figure 1, the kinetic energy of an elemental
slice of bundle p is given by
dT (S) = i i, dÇ • (v a)Y (1)P 2 p s ^ sp }
-6-
where mp is the mass per unit length of the bundle and V
is the lateral velocity of the slice of thickness dÇ.
For small angular deflections along the fuel string
P-1
sp q q(2)
where £ is the length of bundle q and %, is the distance of the
slice from the top of bundle p. (Terms subscripted q = 0 are
zero.)
From equations (1) and (2), the kinetic energy of bundle p
becomes
(8)2
Lq=Oq q
(3)
Evaluation of the integral gives
p p• + I g ' \ 2
+ ± si 2 ' 2
q q 2 p pi 12 p p (A)
The lateral velocity of the midpoint of bundle p is
(5)
Hence equa t ion (4) can be w r i t t e n as
-7-
TP ( S ) =
The two terms on the right-hand side of (6) are the translational
-| T (TRANS> a n d rotational -| T ( R 0 T ) kinetic energies of bundle
p respectively.
For a string of N bundles, the total kinetic energy of the fuel
assembly is
N
As) _ I V % (TRANS) + __ -" 2 2-J P 2
p = l
2.1.1 Matrix Formulation
The translational kinetic energy of the structure is
V ) P (m^p) *(i
P=i p=i V p 4
In matrix notation, equation (8) becomes
T(TRANS) = 1 , „ . [ T . - . 7 . 7 "I
- T - ul T - u•| * y . (m * l)u • y (M4)
(|) (|)
-8-
where o is a (1 x N) row vector of ones. Mid bundle displace-
ment is related to bundle angle by the following matrix equation
•«,, 0
122
r*i
i.e. , y = a() ^
Differentiation of (M5) with respect to time gives
(M5)
= a (M6)
Substituting for y 1 in expression (M4), the translational
<!>kinetic energy becomes
- T.(TRANS) 1 , ' . aT . (- t -)D (M7)
Similarly, the rotational kinetic energy becomes
(ROT) = 1 * £ . [$ * i * râ * £3 * 4>* £
1 ^= ^-*(J) • (~ * m * £. 3) D • j) (M8)
-9-
Adding expressions (M7) and (M8), the total kinetic energy of
the structure can be written in the form
T ( S ) = | * * * T • $ (M9)
where T is a square matrix given by
T = ctT • (m * £ ) D • a + (yy * m * I3)** (M10)
Because kinetic energy is a scalar quantity, the evaluation of
the matrix equation (
numerical value for T
the matrix equation (M9), at any instant in time, gives a single(s)
With reference to the rules for the differentiation of matrix
equations given in Appendix II, from expression (M9) the Lagrangian
terms become
d /3T ( S )\— I — = T • $ (Mil)dt \3q / ^
because T is symmetric, and
= 0 (M12)3q
2.1.2 Computation
The matrix a, in (M6), is calculated in statement DYNMOD 237 as
follows
1 -D
a = ($2 + ~ * I) • I (M13)
Statements DYNMOD 246, 247, and 248 then calculate the total
kinetic energy coefficient matrix T. The column vector M in
-10-
the program contains the bundle masses of the structure,
i.e., M = m * Jl.
2.2 Kinetic Energy of Fluid.
The lateral kinetic energy of fluid attached to bundle p is
(9)
0
where M is the hydrodynamic (added or virtual) mass of fluid jt.P (£•)
unit length associated with the bundle, and V, is the relativeP (E)fluid velocity resolved normal to the bundle. If u is the
longitudinal flow velocity at the slice dÇ, then
fp
3y\
,8t/.+ u
3y(10)
By comparison with equation (2),
fp Z-/ qq=0
p L p Vax/ V 2 p /J
/9y\where I — I for small angular deflections.
Therefore, the fluid kinetic energy for bundle p is
( f ) A,7 j • £'P- l
^ J6 (b t (bi i _ ^ q q iq=0
(11)
-11-
where the hydrodynamic mass and flow velocity are assumed to
vary linearly along the bundle. M and u are the hydrodynai
mass and flow velocity at the mid point of bundle p.
If the fuel assembly is modelled such that the cross section of
each bundle is constant along its length, then, for a uniform
flow channel, - — and -~— -> 0.
Equation (11) simplifies to
v A i f^r - •0
Therefore, the lateral kinetic energy of fluid attached to the
whole fuel assembly is
N N £_ pp-1
V T (f)
P=i
I rp-l -,
/ 2 p 2L-* q vq s vpL Q=0 J
•£ M 2 u
2 p[ p
p I2
p=l 0
Evaluation of the integrals gives
-12-
N
Z r f 2 p p | [ ^ q y q 2 p v p ] 12 pp=l
N - _ !
p P P ' P I / , q rq 2 pP=i
v2
The fluid energy equation is now translated into matrix form.
2.2.1 Matrix Formulation
Because of its similarity to equation (6), the first term in (14)
can be expressed as
T < * > - I * 5 • ( a « (M * £ ) D • a + J LA z 1Z
The second and third terms become
T _ ( f ) = <j) • (M * I * u ) D • a • 0 (M15)
a n d
= - | * (jiT • (M * I * u 2 ) D • $ (M16)
- 1 3 -
w h e r e T ( f ) = T < f ) + T < f > + T ( f ) ( M 1 7 )
A D 0
Subst i tu t ion of (M17) into the relevant terms in Lagranges
equation gives
d f 3 (TA(f) + TB
(f) + Tc( f ))) 3 (TA
(f) + TB(f) + T c
( f ))
dt 3q 3q
Differentiation of the quadratic expressions (M14) and (M16),
and the bilinear expression (M15), leaves the following
Lagrangian terms:
d / 3 T A ( f \ l~ T - - D 1 - ~3 D l "— I — )= a ' (M * Z)u • a + y=- * (M * Z ) • $d t \ 3q ' L <\, °u yl J
(M18)
d / 3 T ( f \ r _ - | T _— [— )= (M * £ * Û) • a • $ (M19)dt \ 3 q / L ^ J
d /9T c ( f >\— — U = 0 (M20)dt \3q /
— = 0 (M21)
3 T * r* T
— B = (M * I * Û ) D • a3q L 'v, J
3T,< I ^ _ _ 11 I "
(M22)3q
i
-14-
~2 D[M * I * u ] • 0 (M23)
Substitution of expressions (M18) to (M23) into the fluid kinetic
energy terms in Lagranges equation gives
= TB , • $ + Twl • $ + Twn ' 0 (M24)d /9T V \ 9T
d t \ 9 q / 9q % ^
where the coefficient matrices on the right-hand side are
T = a T • (M * £ ) D • a + jj * (M * « , 3 ) D (M25)
T . , , = f (M * £ * û ) D • a |T - (M * £ * Û ) D • a ( M 2 6 )
= - (M * I * u 2 ) D (M27)
2.2.2 Computation
The coefficient matrices T Q, T_. and T „ are evaluated by
statements DYNMOD 303, 304 and 30fi respectively.
Ihe column vector VM in DYNMOD is equivalent to (M * £).
Elements of VM are the hydrodynamic masses associated with
each bundle.
-15-
2.3 Gravitational Potential Energy of the Structure
With reference to Figure 1, the potential energy gained by an
elemental slice of bundle p, when the fuel string is displaced
from its equilibrium position» is given by
m dÇ • g • h
The vertical displacement of the slice is
h = 7 H (1 - cos <|> ) + Ç (1 - cos d> ).*^ q q p
Hence the gravitational potential energy of bundle p is
v (G) = / » g V . K è *_2) + 5 è *.') I dç (15)
1 2where for small angles, 1 - cos<(> = — <j) .
For the whole fuel assembly, the gravitational potential energy
becomes
p=l L q=0
An operator Y has been introduced into equation (16) to define
the direction of the gravitational force. If the bundle string
is supported from above, as in Figure 1, then Y = +1. When the
string is supported from below, Y = -1. The effect of gravity
on the dynamics of the fuel string can be ignored by setting
Y = 0.
-16-
2.3.1 Matrix Formulation
The total gravitational potential energy, given by equation (16),
can be expressed as
V ( G )= (-| * Y * g) * [(m * £ ) T • a • 4>2]
= iT * [(4 * ¥• * g) * (m * E) T • a ] D * ï (M27)z ^
Hence
V ( G ) = \ * * T • V_ • $ (M28)
where the gravitational potential energy coefficient matrix
of the structure is given by
Y * g) * (in * l)T • a ) D
= ((Y * g) * (aT • (m * £)))D (M29)
For the articulated bundle string idealization, V_ is a diagonal
matrix.
2.3.2 Computation
\r is evaluated in statement DYNMOD 252.
-17-
2.4 Strain Potential Energy
The strain potential energy stored at tha p t h bundle joint
consists of the following two components:
the interbundle bending strain energy
( C S T )n (i - 4 ,p p P~ *•
(17)
and the end plate bending and rocking strain energy
- Vi>
Shearing of bundle p gives the strain energy component
\ KPAR p
2.4.1 Matrix Formulation
For the whole bundle string, equations (17) and (18) transform
directly into matrix form as
<C«>
(END) = 1 t -T . /J
°(M30)
( M 3 1 )
where B and ^
-18-
Here 3 is the identity matrix with each element on the lowera.
subdiagonal set equal to -1.0.
Similarly for equation (19),
(PAR) _ 1 * -T .J JPAR 11
^ 21 ^PAR 22,
PAR 22
VPAR 12 = VPAR 21
k , k and k are column vectors of stiffness values.Lbi CJJMU r AK
2.4.2 Computation
The matrix 3 is calculated in .statement DYNMOD 239. The strain
potential energy coefficient matrices V ^ , V ^ , V p A R ^ andvD,n n a r e constructed in statements DYNMOD 253, 254, 263 and
264 respectively.
-19-
2 . 5 Structural Damping Forces
Vibration energy can be dissipated by friction at structural
discontinuities and friction within the materials from which
the fuel assembly is constructed. These effects are incorporated
in the dynamic model as structural damping forces.
Structural damping exists when the damping forces are propor-
tional to the elastic forces, but act in a direction opposite
to that of the velocity.
( E ")If the generalized elastic forces acting on bundle p are Q ,
then the corresponding structural damping forces are given by
p (20)
where g is the structural damping factor and i = /-I represents
the n/2 phase difference between velocity and displacement.
2.5.1 Matrix Formulation
In matrix form, equation (20) becomes
Q ( D ) = i * i * Q ( E ) (M33)
(ST)Because the strain potential energy of the bundle string V
is derived from generalized elastic force terms, the following
relation applies
3V ( S T )
Q(E)= - — (M34)
Combination of expressions (M33) and (M34) gives
-20-
(M35)3V ( S T )
Taking (M35) to the left-hand side of equation (M2), the
Lagrangian potential energy term becomes
8V 3V(G> 3V ( S T )
— = + (Ï + i * g) * — (M36)3q 3q 3q
This formulation is satisfactory for steady harmonic oscillations
of the bundle string where the velocity at any point will be IT/2
radians out of phase with the displacement. However, in the
case of a transient forced response or fluid elastic instability
the velocity and displacement are no longer in quadrature,
(e.g., With buckling of the fuel string, the velocity and
displacement are in phase.)
If the velocity leads the displacement by a phase angle 0, then
the term given by (M36) must be adjusted to the following form
3V 3V(G) _ _ 3V ( S T )
— = — + (Ï + (cos6 +i*sin6) * g) * — (M37)3q 3q 3q
In most transient response problems G will be a function of time.
Therefore, a knowledge of 0(t) will be required for the accurate
computation of transient response with structural damping. This
might involve lengthy iterative procedures and has been avoided
for the present. The steady harmonic term (M36) is used in the
dynamic model.
-21-
Three column vectors of structural damping factors are used in
the model. These are 8CST» S^ND a n d ^PAR c o r r e sP o n d : L n8 to
interbundle (central support tube) damping, end plate damping,
and bundle parallelogramming damping respectively. The p t n
element of each damping vector corresponds to a damping factor
peculiar to bundle p.
With structural damping, Lagrange's equation will contain the
following three generalized elastic force terms,
1 + i * g"
Ï + i * i
CST1 3V
/ * ~CST
(CST)
3q(M38)
c 'END
1 + i * gEND
3v(END)
3q(M39)
and 1 + i'PAR
'PAR
where the (2N x 1) column vector
equivalent to 1 t i g]
i -i- i gf1 + i g,
1 + i gN
9V(PAR)
/Î + i * i\
\ï + i * £/
(M40)
is
,<ST)
The general expression (1 + i * g) * — in (M36) is equal3q
to the sum of terms (M38), (M39) and (M40). Differentiation of
-22-
the matrix expressions (M30), (M31) and (M32) allows the
Lagrangian potential energy term to be formulated, i.e.,
/I + i * i C S T\ av<CST>
\l + i * -g* — = (V.OT + i * g.c-r * V,CT) •• (M41)
3q
(END)
* VBlin) •* (M42)1 + ± * ^ N D 7 3 q
\ï + i * ë. 3q
- - D - •
1 9 \ / v ePAP PAR 6PATJ "PAR 1
21 %PAR 22
(
^ (l*8PAR*^AR) (l*8PAR*^AR) '.
(M43)
Addition of (M41), (M42) and (M43), together with the derivative
of the gravitational potential energy
3V ( G )
— = V • <t> (M44)3q ^
gives the Lagrangian term (M36). This can be expressed as
3v / v 1 ; L v 1 2 \q (M45)
-2 3-
where the coefficient matrices are
111 " iG + ICST + IVAK 11 + (1 * 5cST * *CST> + (± * «PAR
12 " (i * AR * KPAR)
+ V + (i * sL * V ) + (i * ë * ÏL )*PAR 22 U ^ND ^END; vx ^AR
An alternative method of incorporating the structural damping
forces in the model is presented in Appendix III. This ascribes
a damping factor to each string (as opposed to each bundle) in
the idealized fuel assembly.
2.5.2 Computation
The structural damping factor column vectors IMGCST = i * g-,OTLi O X
and IMGEND = i * i F N n are calculated in statements DYNMOD 2 76
and 277.
After forming the diagonal matrix DKPARD = (i * iD.D * KDAI()r AK r AK
in DYNMOD 278, the potential energy coefficient matrices V.,,
V „, V and V„„ are constructed in statements DYNMOD 279, 28l
and 28?..
The square matrices SDCSTX and SDENDX, in DYNMOD 2 79 and 282
respectively, contain structural damping terms associated with
the constraints applied to the ends of the fuel assembly (see
section A.3.1) .
- 2 4 -
2 . 6 Hydrodynamic Forces
At this stage, the hydrodynamic forces to be incorporated in
Lagrange's equation consist of those forces which can be
expressed as functions of the generalized coordinates or. their
time derivatives. Analytical expressions describing these
forces are presented below.
The locations and directions of action of the hydrodynamic
forces are shown in Figure 1.
2.6.1 Longitudinal Friction Forces
Dimensional analysis, for a cylinder in axial flow, suggests
that the longitudinal friction force per unit length is "i < ••
by
I C F P "2 D
where D is the cylinder diameter,
p is the fluid density,
u is the flow velocity,
and CF is a coefficient of friction (ref. 2)
Therefore, the longitudinal friction forces per unit length
acting on a bundle of n fuel elements of diameter D will be
1 2
F = — pnDu C_, Cosij)L 2 F
where <\> i s the angle between the bundle axis and the flow
d i r e c t i o n . For small angles Cos<j> -»• 1, hence
1 2
FL = - pnDu Cp (21)
-25-
2.6.2 Normal Viscous Forces
In deriving the expressions for the normal viscous forces,
it is assumed that no separation occurs in cross-flow.
There are two components of viscous hydrodynamic force normal
to the bundle string. One is the drag force per unit length
and is given by
- pnD C —2 D 3t
where is the lateral velocity of the bundle slice. C
TE" D
is a viscous drag coefficient at very low relative (bundle
to fluid) velocities; it has the dimensions of velocity. The
viscous force given by the term (22) acts in opposition to
the motion of the bundle element and acts as a damping force
even in the absence of fluid flow.
The other viscous force is the normal component of the friction
force. This is defined as
where F is given by equation (21) and v is the normal velo-
city of fluid relative to the bundle. From Figure 1 it can be
seen that
Hy 3y 3yv = + u sin <f> = + ur dt 3t 3x
Therefore the normal component of the friction force per unit
length becomes (refs. 3 and A)
3y(23)
3t
-26-
Addition of terms (22) and (23) gives the normal viscous
force per unit length
/ 3y 1 3y \ 1 3yF - F — + 1+ - pnD C — (24)
N L \ 3x u 3t / 2 D at
2.6.3 Longitudinal Pressure Drop Forces
Bernoulli 's energy equation, for fr ict ionless incompressible flow, is
u2 P_ _ _ gx = a constant, where P is the2 P
pressure in the fluid. Differentiation with respect to x
gives3P 3u— = - pu — + pg3x 3x
Therefore, the longitudinal force per unit length caused by
hydrostatic and velocity head pressure drops is
3? 3u
A — = Apg - Apu — (25)9x 3x
where A is the cross-sectional area of the relevant bundle.
Frictional forces along the cylinder also contribute to the
pressure drop. The friction pressure drop is caused by fluid
shear forces at the surfaces of the flow channel and fuel
assembly. If Du is the hydraulic diameter at the point ofn
interest the friction pressure drop force per unit length is
given by
2- pnDu Cv — (26)2 DZ H
- 2 7 -
Hence the t o t a l pressure drop force per unit length i s
8P 3u 1 D2
A — = YApg - Apu — n C pu (27)9x 3x 2 Du
n
where the operator Y was defined in section 2.3.
2.6.4 Lateral Pressure Forces
The lateral pressure drop force per unit length acting on
bundle p is given by (ref. 4)
= A(P) [ — F1" I — I (28)
From equation (27), (28) becomes
= YApg - Apu n C^pu —py
In the model P(x) i s assumed to be a l i n e a r function of x
over the length of a bundle.
2.6.5 Base Drag Force
The long i tud ina l base drag (form drag) force ac t ing at the
free end of the bundle s t r i n g i s represented by the empir ica l
equation ( ref . 1)
1 2 2F_ = p , D u , C_ (30)
B — end eq end B
2
where p , and u , are the density and velocity of fluid at
the end of the bundle string, and D is the equivalent diameter
of the free end. C_ is the base drag coefficient.o
- 2 8 -
2.6.6 Non-conservative End Force
A non-conservative lateral force F acts at the end of theN 0
bundle string as shown in Figure 1. This force accounts for
the loss in lateral momentum flux of the fluid at the free
end. It is a function of the shape of the nose or tail section,
The force is given by the expression (ref. 1)
F N C " ^ " " e n d " e n d I I » I , ^ ~ \~T I A\ \ 8x / e n d u , \ 9 t / e n de n d
which reduces to
F.T_ = - (1-f) M u [( — ) + u <f> \ (31)NC end endlx ». / end N "
M , is the hydrodynamic mass per unit length at the free endend
of the bundle string.
f is a shape factor for the free end and can assume a value
0<f<l. If the free end of the string is streamlined, f
approaches unity and the restoring force F _ becomes negligible,
For a blunt end f goes to zero.
-29-
2.7 General ized Moments
The hydrodynamic forces per uni t length given by equat ions
(21) , ( 24 ) , (27) and (29) , and the forces ac t ing at the end
of the bundle s t r i n g given by (30) and (31) can now be t r a n s -— (H)
formed i n t o the genera l ized moments Q in Lagrange 's equat ion
(M2). The genera l i zed moments are most e a s i l y formulated by
ufing the principle of virtual work.
If ôoj is the work done by the hydrodynamic forces v/hen bundle p
is rotated through the infinitesimal angle 6(|> , then the
generalized moment associated with this virtual movement is
given by
With re fe rence to Figure 2 , the work done by a l l the forces
along the s t r i n g in d i s p l a c i n g bundle 1 by 6<j> i s given by
dÇ - / -1 F ^ Ç ^ from forces on the first bundle
AT W -/VJ F ^ ^ ^ - ^ .
/ / 3
/ J Fpy3> £i6<t)id5 7 FN
from forces on
the second
bundle
from forces
on the third
bundle
S work done by forces acting on the other bundles!
- 3 0 -
S PThe term [A (•£=-) l ] £., 60. (cf) -<i ) is the work done by
p ox p p 1 1 p i
the pressure drop force acting at the end of bundle p. In
the construction of the expression for 6u)_ , the following
small angle approximations were made
s i n |> -4>-.
a n d c o s (J) = c o s (<j)P P
>n )1
1 .
The work done by the fluid forces in rotating bundle p by
an amount Sd> can be found in a similar fashion.P
Using r e l a t i o n (32) , the general ized forces associa ted with
the general ized coordinate cf> are found to be
£ £<»» . /
i=p+l
/*>
+ J F r ( i ) l ( * . - • ) d ç - F A . <•§£). 1.1 £ ( 0 . -^ L p i P I l 3x l i l p T i
FNC ( 3 3 >
for p = 1, 2 N.
The generalized forces associated with bundle shear are as s une d(VI)
to be n e g l i g i b l e , i . e . , Q ' •+ 0, for p = N + 1 2N.
-31-
2.7.1 Matrix Formulation
The terms in equation (33) can be divided into three groups.
These are (i) the terms containing F , which is a function
of the variable E,,
(ii) terms containing forces F and FT , whichpy L
are assumed constant along each bundle,
and (iii) those terms containing FD and F , the forces
at the end of the bundle string.
Adding terms within each of the above groups gives the
generalized moments Q , Q and Q respectively, where
j2 ri-p+i J
v o
p p p p
a. Evaluate Q
From equation (33)I N ?,.
and from equation (24)
| pnD CFu + \ pnD C,,) | f + ^ | j (36)
If H = — pnD C u + — pnD C , where H is a constant for each
bundle, then equation (36) becomes
Therefore, the force per unit length at the slice dÇ, a
distance Ç from the top of bundle p, will be
- 3 2 -
(5)
However,
there fore
4*>9t p
i, d) + Ç<J) (see equation (2))q q P
q=o
. p-i
q=o
Substituting (38) into the first term of (35) gives
(38)
q=o
After performing the integration, this equation becomes
p-.l
Similarly, the second term in (35) becomes
N £.
Tq pK + F<«1 Ij
q=o
.S.
p - 1
i
2
( i )
i=p+l q=o
(40)
- 3 3 -
H e n c e , combin ing e q u a t i o n s (39) and (40)
> £ , ) , + H < t . + F T( p )
P 2 ^ - — ' q q P 3 P L 2
q=o
p ~ 1
i=p+l q=o
In matr ix n o t a t i o n , equat ion (41) becomes
Q(h) . H° . R . t
HD . H° . Si(M46)
where 8 is a square matrix with ones above the diagonal andO/ _
zero's elsewhere. F is a column vector of constants and isi-i
given by
FT = iL Z
(M4 7)
Expression (M46) can be rearranged into the form
(M4 8)
- 3 4 -
The coefficient matrices are
Q = - ID . é * H * 1)D . a , , , . . + 9 . (H * 1)D . a ] (M49)
and
(M50)
where
ï(2/3) = ? • lD + f * l ° (M51)
b. Evaluate
The pressure drop moment and longitudinal friction moment
terms in equation (33) are now examined.
Collecting the terms together gives
(z) fp (P) V ^ C
p J py / J j py po i=p+l o
N I N
+/
P (±) X " 3P
F, £(<)>. - ()) )dC - 7 (ATT") I t (* - è ) (42)I. p i p • / ^ 3x l l p l pi=p+l ° i=p+l
A f t e r i n t e g r a t i o n , (42) becomes
F ( ) i y ; F( i > .
py \ , / p x - ^ py iN l / i=P+i
*A9x • ^i~® ) ? - ( A 3 )
T^p+1 i=p+l
- 3 5 -
where F and FT are assumed constant over each bundle.py L
Be cause
Fp y V 3 x / \ d * J [ d x l V
equation (43) can be rearranged to give
P P 2
N
i=p+l
+ A ? I F ( i ) 4»4 (44)P l—â P L T i
i=p+l
In matrix notat ion (44) becomes
Q(Z) - I • ft
* 9 • £ * F L * (J) | (M52)
Hence
where
Q ( Z ) = Q-o • • (M53)
Q. = D • I I £ *
(M54)
- 3 6 -
3PThe column vector of constants (A-g—) is given by
A-|£) = Y * g * p * Â - P * Â * û * ( |^) - | * p * li * D * ? * CF * D T DH (M55)
(e)c. Evaluate Q
The moments associated with the hydrodynamic forces at the end
of the fuel assembly are
= ^ p , D 2 u 2 , c I (<t>-<|> )2 end eq end B p N p
- (1-f) M , u , ( ( I 2 ) , + u , <fu Iend end I dt end end Nî
3 N
From e q u a t i o n ( 2 ) , (~t\) • = / . & d> » h e n c ed t e n d ~. q q
N
-B I > ^q^q + Uend
q=J
where
eA = \ Pend °eq Uend CB
e and e a re s c a l a r c o n s t a n t s .A o
Equation (46) can be rearranged to give
N(e)
(45)
a n d eB = _ ( i _ f ) Mend u e n d (48)
(e) V \ *Q = £ (eA + e_ u ) $i - l e à + I & > I d> (49)
P p A B end N p A p p B / J qvq v 'q=l
- 3 7 -
This equation now transforms into matrix notation to give
Q ( e ) = QE1 • I + QE0 • $ (M56)
where
* 1 * (Ï1 * ONES) (M57)< \ , E 1 B
a n d
[ "ID(e. + en * u ,) * 1 \ . zl - (e. * I) (M58)A B end J ^ A
11 isONE? i s a s q u a r e m a t r i x of o n e s , i . e . , / -,-, • \
and. zl is a matrix of zero's and ones such that'X,
/0 0 0....1, / 0 0 0....
i [i ; ;\o 6 6
Collecting together the coefficient matrices, the generalized
moment equation becomes
y • •+ ( > + ;EO
+ y
2.7.2 Computation
The generalized moment coefficient matrices, derived from the
hydrodynamic forces, are constructed in statements DYNMOD 325 to
34l:.
After evaluation of the column vectors H and F , and theL
square matrix a,n... in statements DYNMOD 334 to 337, Q u n andQul are formed in DYNMOD 339 and 341.H 1
-38-
Before forming Q _, the cross-sectional area A and pressureZU r\ -y»
drop force per unit length (ATT— ) are calculated for each
bundle (DYNMOD 346 and 3A7). Q Q is calculated in DYNMOD 349,'X,
In the statement DYNMOD 347, the function DBYDX(u) calculates
the value -jp-, the averagi
bundle, see Appendix IV.
the value -5—, the average flow velocity gradient along a
The flow velocity over a bundle is assumed constant. However,
because adjacent bundles can have different flow velocities,
some account must be taken of the velocity-related pressure
drop forces between the bundles. An approxi.jiat'.jii for tins
is made by ascribing an average velocity gradient to the --r'
from which a mean pressure drop gradient can be calculât
The constants e. and e_, defined by equations (47) and (48)
are calculated in statements DYNMOD 325 and 327. The st - T
function ATEND(k), used in the calculation of e. and e_, ev> ,
ates the value of k at the end of the bundle string, i.e.,
kend1 / 3 k \
N + 2 *N 7") '\3x/N
Q _ and Q,,.,, the generalized moment coefficients describing
the forces at the end of the bundle string, are given by
statements DYNMOD 328 and 330.
-39-
V.qua_tions o f Mo tion
Thf terms in Lagrange's equation have now been evaluated as
functions of the generalized coordinates q and the time
derivatives q and q.
.8.1 Ma_t_ri_x_ Fo rmulation
Substitution of equations (Mil), (M12), (M24), (45) and (59)
into Lagrange's equation (M2) gives the matrix equation of
motion of the fuel assembly,
i.e.,T0/
0y
T
0
0%0
\
1
6•X
0
q +12
q +
'(QEI + V °
22
rTFi °
Ko oT., %
01,
The f ii'ot square bracket on the left-hand side of the equation
contains structural kinetic and potential energy terms
(including the structural damping moments). The terms in the
second bracket describe the fluid kinetic energy. On the
right-hand side of the equation are the generalized moments
caused by the hydrodynamic forces acting on the structure.
-40-
Rc ar rangc.inen I of the equation of motion gives
'(T + T ) 0 \ _ /(T - Q - Q ) 0a, -J2 -u \ . q + [ ',F1 %E1 <vH1 % \ . q
i(\T + T — 0 — 0 — 0 ) VWL1 FO WEO h( 4z0; V120 (M60)
Pfirtial matrix multiplication in (M60) yields the two expres-
sions
(Ï
QE0 " ?H0 " JiO* • * + Jl2 • * " 5 (M61)
andV?l . 1 + V 7 2 . * = 0 (M62)
' r o m u 1 t i p 1 y i n R ( M'i 2 ) h y V leave .-3
Ï = - V 2 2 ' . V 2, • * (M63)
S..'. •• 'i ti i n c :i on of (M63) into (M61) forms the differential equation
:•• • me ' 'on (M31 of the fueJ assembly
A • ; + B • i, + C • ? = 5 (M3)
-41-
where
A = T + T ,F 2
j 'u ^j ^J Mj Hj '\j ^ j l*\j
Computation
The coefficient matrices in the equation of motion are
assembled in statements DYNMOD 364, 368 and 372. Matrix V
in DYNMOD 372, the effective potential energy coefficient
matrix, is constructed in DYNMOD 293.
The generalized coordinate transform matrix V , which relates
the two sets of angular deflection coordinates, is evaluated
in DYNMOD 2 84,
i.e. From equation (M63),
i?> = V R . 4> ( M 6 4 )
hence ,
InTe^er function TRIINV is used in the calculation of V%R
to invert the tridiagonal matrix V . If V is singular;i message is printed.
'„ is used in DYNMOD 293 to calculate VK ^
-42-
2.9 Solution of the Equations of Motion
Because of the fluid and structural damping terms, matrices
B and C in (M3) are not symmetric. Therefore, the matrix
equation of motion represents a coupled set of equations.
2.9.1 Matrix Formulation
Expression (M3) is solved by reducing it to a first order
differential equation given by
BR . z + ER . z = Ô (K65)
where BR =/ ^ I , ER
(M65) is solved by assuming harmonic solutions of the form
5(x,t) = V(x) * exp(iut) (M66)
where exp(iyt) is a scalar variable. Substitution for z and
z in (M65) gives
BR . (iu*V) + ER . V = 5.
Pre-multiplication by BR and subsequent scalar multiplication
by i leaves the eigenvalue problem
(i * BR"1 . ER - u * I) . V = Ô (M67)
where I is the identity matrix. Solution of (M67) will give
2N roots for u, the eigenvalues; the natural frequencies of
the fuel assembly are given by U/2TT.
-43-
The mode shape of the bundle string, at a natural frequency
((J/2TT) . will be described by $., which is the angular dis-
placement vector part of the eigenvector V. =(j).•
Horizontal joining of all the vectors V., in spectral order,
provides the modal matrix V. For each modal matrix of bundle
angular deflections $, there is a corresponding matrix of
endplate deflections ^ given by
'i' = (-V I 1 . v,.) . $ (M68)
The lateral deflection of the bundle string in the j mode
is given by
Yr = CMAT . $ (M69)
where CMAT is a coordinate transform matrix.
In general, as matrix B is non-zero and is not proportional
to A or C, each element of an eigenvector will be characterized
by its relative phase in addition to its relative amplitude.
Therefore 2N equations will be required to determine all
elements in each mode of oscillation of an N degree of free-
dom sys tem.
The solutions <j. of equation (M3) are normalized vectors.
They are not absolute values of bundle angular deflections.
-44-
2.9.2 Computation
The inverse of matrix BR is calculated in statement DYNMOD
392 and matrix ER is constructed in DYNMOD 393.
After forming the matrix W in DYNMOD 394 where
W = i * (BR"1 • ER) , (M70)
the equations of motion are solved using the LAMPS integer
function MODAL in DYNMOD 395.
If the idealized fuel assembly has N degrees of freedom in
coordinates <J>, then matrices
matrices of order (2N x 2N) .
coordinates <j>, then matrices BR , ER and W are all squarea. a. ^
In DYNMOD 395, Mu (= \i) is a column vector of 2N eigenvalues,
and MODES (= V) is a square (2N x 2N) matrix of eigenvectors."V.
The natural frequencies FREQ(E u* ) of the fuel/fluid system2TT
are calculated in DYNMOD 401. The lower half of the modal
matrix MODES contains the eigenvectors of bundle angular
deflections. These are extracted and normalized to give
the (N x 2N) matrix PHIS in DYNMOD 402.
Using the coordinate transform matrix CMAT, the lateral^ r
deflections of the bundle ends, DEFLEC (EY ) , are'Xj
calculated in DYNMOD 404, Bundle endplate angles corres-
ponding to the matrix PHIS are evaluated in DYNMOD 405.
- 4 5 -
3. INTERPRETATION OF STABILITY RESULTS
For a fuel assembly idea l i zed in to N bundles there w i l l be 2N
eigenvalues and 2N e igenvec to r s . A t y p i c a l output i s shown
in Figure 3 . The j n a t u r a l frequency (y/2ir) . correspondsth J
to the j vector of normalized l a t e r a l bundle end de f l ec t ions
Y. (x) . Hence the j so lu t ion of the homogeneous equations
of motion (given by M65) i s of the form
yV. ( x , t ) = Y r(x) * exp ( i u . t ) (M71)
3. I Ei genvalues
Usually the j t h eigenvalue takes the complex form
u. = y.R + i u j j (50)
and the equation of motion of the assembly for the j mode
can be wr i t t en as
! ( x , t ) = Y rv ' x ) * I e x p ( - p t ) . e x p ( i y t )
1 .1 |_ 3l J K J(M72)
The na tu ra l frenuency of v ib ra t ion of the assembly for t h i s mode
i s given by IJ / 2 TT . As t s t e a d i l v increases the term exp ( i lJ ._ t )1 R i R
i n d i c a t e s tha t points along the assembly w i l l experience simple
harmonic motion. However, the magnitude of the term exp(-u t)
increases or decreases with time. If the imaginary pa r t of the
eigenvalue i s p o s i t i v e , v . ( x , t ) decreases exponent ia l ly with timet" h
and the fuel assembly will bs damped in the j mode. The
damping is given by ( j/'-O*
Alternatively, when u. •, is negative, the vibration amplitude
of the assembly increases with time. This is an unstable
condition and is referred to as a fluidelastic instability.
-46-
Under certain circumstances u goes to zero and the natural] K
frequency is purely imaginary. This corresponds to a degeneratemode and indicates critical damping or bucklin_g of the fuel string.
3.2 Eigenvectors
Like the eigenvalues, the eigenvector elements can be complex
numbers. For example, the normalized displacement at the base
of bundle p in the j mode might be
JP(51)
From equation (M71), the motion at this location is described
by
ip(t) = ( r . exp i
(52)
In general, the relative phase angles
,r
t an
at the ends of adjacent bundles are different. Therefore the
complex eigenvector describes a non-stationarv mode of vibration
and the motion will often resemble a wave travelling along the
fuel assembly. For simnlicitv in all further discussions, "non-
stationarv vibration modes" will be referred to as "travelling
waves". In the absence of structural damping and hvdrodvnamic
forces, the bundle ends will vibrate in nhase or antiohase for
all modes. Here, the travelling waves have become standing waves
- 4 7 -
3 . 's Modrxl P a i r s
When th>3 f u e l a s s e m b l y i s s t a b l e i n a l l i t s m o d e s a n d s t r u c t u r a l
d a m p i n g i s n e g l i g i b l e , t h e n t h e n a t u r a l a n g u l a r f r e q u e n c i e s a n ;
p r i n t e d i n p a i r s o f t h e f o r m
y . = y + i u .3 JR J
a n d p j + 1 = - U . R
w h e r e j i s n o w a n o d d i n t e g e r . ( M u l t i p l i c a t i o n o f e q u a t i o n s
( 5 3 ) b y i g i v e s a c o m p l e x c o n j u g a t e p a i e o f n u m b e r s . ) T h e m
a r e N p a i r s o f f r e q u e n c i e s f o r a n N d e g r e e o f f r e e d o m S y s t e m .
I f t h e f u e l a s s e m b l y e x p e r i e n c e s a p u r e f l u i d e l a s t i c i n s t a b i l i t y ,
a g a i n w i t h o u t s t r u c t u r a l d a m p i n g , t h e n t h e r e l a t i o n g i v e n b y
e q u a t i o n s ( 5 3 ) s t i l l a p p l i e s .
H o w e v e r f o r bu c k l i n g - t y p e i n s t a b i l i t i e s , w h e r e ;i v 0 , t h e n
u i . e . t h e i m a g i n a r y p a r t s n r e n o l o n g e r c q u : - i l .
F o r e a c h p . i i r o f n a t u r a l f r e q u e n c i e s t h e r e w i l l b e a e n r r c s -
D o n d i n g n a i r o f b u n d l e d e f l e c t i o n e i g e n v e c t o r s . T h e p a i r s o f
e i g e n v.i l u e s a n d e i g e n v e c t o r s a r e k n o w n a s m o d a l p a i r s . T h o s e
d e s c r i b e t h e n a t u r e o f t h e w a v e s t h a t t r a v e l i n e a c h d i r e c t i o n
a l o n g a p e r t u r b e d f u e l a s s e m b l y .
I f s i g n i f i c a n t s t r u c t u r a l d a m p i n g i s p r e s e n t i n t h e f u e l
a s s e m b l y a n d i t s s u p p o r t s , t h e n e a c h m o d a l p a i r o f f r c i n i
t a k e s t h e f o r m
u . = u . „ + i n .
V7h.?ro i i s a g a i n an o d d i n t e g e r . H e r e , p ^ |i . , a n d] K ( j + J ; K
-48-
Consider first the real parts of equations (54). These describe
the natural angular vibration frequencies of the assembly. If
Ay., is defined as (11, , - y._) then as the flow rate increasesJ K (. J+-LI K ]K
the function An /p also increases. i.e., t'he vibrationJ R 3 K
frequencies of waves travelling in opposite directions alongthe fuel are different.
The frequency dispersion, for a particular modal pair, suggests
that the unidirectional flow encourages waves to travel faster
in one direction than the other. This phenomenon is equivalent
to the Doppler frequency shift of classical wave theory. An
example of this effect is shown in Figure 4.
A little dispersion exists at zero flow rate if the bundle string
is not uniform along its length and structural damping terms
are included in the model. This effect is analogous to wave
propagation in anisotropic media. (In experimental investiga-
tions of fuel string response to axial flow forces, large
amplitude low frequency vibrations are sometimes observed.
These might be caused by beating between waves of a modal pair
travelling up and down the assembly.)
The imaginary parts of equations (54) are each constructed
from the two terms S and V where
u = V + S
and UfiJ.i\r = V - S (55)
S is the contribution to the damping from structural damping
forces, and V is the contribution to the damping or excitation
from the hydrodynamic forces. For a given modal pair, addition
and subtraction of equations (55) will resolve V and S
respectively. When examining the dvnamic stability characteristics
and total damping of a structure, the imaeinarv term u._ = V+S
mus t be used .
- 4 9 -
For al l stable modes, pure ins tab i l i t i es , and modes that have
returned to the stable state at high flow rates (see Figure 36),
V and S can be separated as described above. In these cases
the damping ratios of waves in the string are y /y and
^ / ! J Where V = V + S
Occasionally, for a particular eigenvalue, the condition
[ l j | < < | v i j exis ts , where y._ 4- 0. This suggests that the
damped or excited fuel assembly is oscillating with a very
low frequency. Examination of the relative phase between velo-
city and displacement along the structure (using the eigen-
vectors V (x) in M67) indicates the nature of the motion. If
the velocity and displacement are in phase, then buckling
has occurred; if the velocity and displacement are in quadrature,
or thereabouts, then there is damped motion or fluidelastic
instabil i ty depending upon the sign of p. .
Results from DYNMOD show that an increase in the viscous damping
of a structure is accompanied by a slight downward shift in i t s
natural frequencies; see Figure 5. Conversely, an increase
in the structural damping produces an increase in the natural
frequencies; see Figure 6. Because structural damping has been
introduced into the dynamic model as an imaginary stiffness, the
overall stiffness magnitude increases with increasing structural
damping. As a result the natural frequencies of the structure
shift upwards. These observations are in agreement with
basic vibration theory.
-50-
4. CONSTRAINED FUEL ASSEMBLY
The dynamic model has been developed by assuming that one end
of the idealized fuel assembly is anchored to a rigid support.
However, for certain designs of nuclear reactor, the fuel
assembly is constrained at more than one position.
By using subroutine PINNED in DYNMOD it is possible to model
fuel strings that are:
a) constrained at the end of the last bundle, and/or
b) pinned at the end of any intermediate bundle.
Examples of fuel assemblies experiencing more than one constrain!
might be those of the British SGHWR which is constrained at
both ends, or those fuel assemblies which, for any reason,
vibrate in contact with their flow channels.
Subroutine PINNED performs two main functions:
(i) It calculates a coordinate reduction matrix R which
defines a new set of bundle angle coordinates <j> ' , where
$ = R . $' (M73)
Application of constraints to an idealized bundle string
has the effect of reducing the degrees of freedom of
the structure. In the dynamic model, the number of degrees of
freedom lost is equal to the number of additional constraints
acting on the bundle string. If there are two extra con-
straints, <j> ' will contain (N-2) elements for a fuel string
having N bundles; matrix R therefore is of order (NxN-2).
(ii) Additional stiffness and structural damping values,
applicable to constraints acting at the ends of the
assembly, are calculated. Therefore, any damped end condition
can be modelled, eg. pinned, stiff, clamped, etc.
- 5 1 -
4 . 1 Coordinate reduction matrix, R
Consider a bundle string whose end is pinned at its equilibrium
position as shown in Figure 7. It is assumed that the pinned
joint is free to move along the x axis.
The displacement y., . can be defined as
N-l
yN-l = V Np=l
i <t> , for small angles
N-l
H e n c e tj> -^ £P=I p p
(56)
Because <pK can be defined in terms of the other bundle angles,
a new generalized coordinate system <jj ' (i.e., <(>' , <j>« ...((>' .) can
be introduced.
From expression (M73) it can be seen that the reduction matrix R
can be defined as
R(N)
MNxN-1)(M74)
-52-
(The subscript after R defines the order of the matrix.)
Similarly, if the end of bundle J is pinned, (see Figure 8),
(J)
(NxN-1)
(M75)
where 4> T = - —J l
J-l
p =(57)
Fina l ly , if the end of bundle J and the end of the s t r i n g are
constrained (refer to Figure 9 ) ,
R(J,N)
MNxN-2)I AM | | BZ 1
AZ fBM
(M76)
- 53 -
where AM (JxJ-1)
(M77)
AZ (N-JxJ-l)=
BZ (JxN-J-l)=
and BM(N-JxN-J-1)
- 1whe re
0
0
0
0
0
0
0
0
0
0 0
. .0
0 0 0
0 0 0
0 0 0
0. : .o
(M78)
(M79)
(M80)
- 5 4 -
4 .2 A p p l i c a t i o n of Matr ix R
4 . 2 . 1 Theory and Matr ix Formula t ion
In the homogeneous mat r ix e q u a t i o n of motion
A-(j> + B-(j> + C-<j> = 0 (M3)
the coefficient matrices A, B and C are all of order (NxN)
for a N bundle string. If the string is constrained at more
than one point, then a new matrix equation of motion, in
the generalized coordinates $', must be formulated.
The method used to reduce the N equations in (M3) to the (N-l)
or (N-2) equations in the new coordinates <J> ' will now be
demonstrated by examining the origins of matrices A, B and C.
Consider the first term in (M3).
A is the sum of the structural kinetic inergy coefficient
matrix T and a matrix T__ derived from the lateral kinetic
energy of the fluid. The generalized structural inertia
forces T • <)> were obtained from the Lagrangian term
(s)d / 3 T ( S ) \
in the coordinate system <J>, where
( c, •) T T 7
Tv = - $ • T • $ (M9)2 ^
(s )T can be converted into a function of the coordinates
$' as follows.
-55-
Differentiation of (M73) with respect to time gives
= R . (M81)
where R is a constant.a,
S u b s t i t u t i n g f o r $ i n ( M 9 ) ,
. . 1 -T()
whe re
2T
= r
. T . R ] . 4"1 (M82)
(M83)
Hence Che Lagrangian term
d
dt
, ( s ) '
in the reduced coordinates becomes [R . T - RJ . <|> ' .
TF R T R1^ ' -., " is a reduced square coefficient matrix.
This approach is used to reduce the order of all the coefficient
matrices in equation (M3).
In general, the reduced set of equations of motion can
be obtained by the substitution of equation (M73), and its
time derivatives, into (M3), i.e.
A . R . v ' + B . R . < t > ' + C . R . <f> ' = 0 (M84)
w h e r e <j>= Jjt.. <j> '
-56-
TPre-multiplication of (M84) by R leaves the reduced equation
[RT.A.R ].£' + [RT.B.Rj. <£' + [R T.C.R]. (f)1 = 0 (M85)
This expression describes the (N-m) equations of motion of
the fuel assembly, where m is the number of additional con-
straints. The reduced eigenvectors $' of (M85) are converted
back into the bundle angle solutions $ using equation (M73).
4.2.2 Computation
a) The coordinate reduction matrix R is constructed in
subroutine PINNED. This provides four calculation option-.
These are: (i) no extra constraints,
(ii) end of bundle string constrained,
(iii) intermediate bundle pinned,
and (iv) both intermediate and end bundles constrained.
As a computational example, consider the fourth case.
Submatrices AM, AZ , BZ and BM are evaluated in statements% -VF % ' %
PINNED 68, 71, 75 and 76 respectively. Integer JP in
these statements is equivalent to J in equations (M76) to
(M80). Matrix Q is assembled from these submatrices in
PINNED 78.
b) R is used to reduce the order or the matrix equation of
motion in statements DYNMOD 37f? to 380.
The logical operator BOTPIN in these statements is set TRUE
if the end bundle of the string is constrained. Operator
MIDPIN is set TRUE if any other bundle is constrained.
-57-
After calculation of the eigenvectors $' of the reduced
equation (M85), the relative bundle angles $ are determined'X,
by pre-multiplying by R. This operation is carried out in
DYNMOD 403.
The relative bundle angles (PHIS) can be converted into
relative bundle displacements (DEFLEC) using a further
coordinate transform matrix CMAT (see section 5.1.2).
DEFLEC is calculated in DYNMOD 404. The natural frequencies
and normalized fuel assembly deflections are then printed
and/or plotted.
4.3 Stiffness and Extra Damping at the End Constraints
4.3.1 Theory and Matrix Construction
The spring stiffnesses K and K describe the constraint
at the first bundle. These are assumed to be dependent upon
the nature of the fuel assembly support. The parallelogramming
stiffness K p A R 1 i s assumed to be a unique property of the
first bundle, and therefore independent of the constraint.
If the end of bundle N is anchored, two additional constraint
stiffnesses can be introduced into the model via subroutine
PINNED. These are KCBOT and KEBOT; they correspond to the
central support tube stiffness and the end plate stiffness
at the end of bundle N respectively.
After being transferred from PINNED to the main program,
KCBOT and KEBOT are added to the strain potential energy
coefficient matrices V___ and V_.,_. KCBOT is added toO.C S T -vE N D
element (N,N) of V and KESOT is added to element (N,N)
- 5 8 -
These additions are jus t i f i ed by an examination of the
generalized central support tube s t r a in moments associated
with the f i r s t and las t bundles,
r(CST)
i . e . f I = (%•„„„., + K,,^,, )<()., - Korlm0 <j),, (58)
a n d
r(CST),
(KCSTN(59)
In a symmetrical structure, for example a uniform beam,
equations (58) and (59) are equal, i.e., K C S T = KC S T 2 '
KCBOT = K , etc.
The new t r i d iagona l m a t r i c e s V and V are
>.CST
( KCST1+ KCST2 ) KCST2
KCST2 ( KCST2+ KCST3 ) " KCST3
-KCST3
( KCSTN (M86)
-59-
and similarly,
END
END1 END2; END2
~KEND3
.". .0
~ KEND3
-KENDN (KENDN+KEB0T>(M87)
Structural damping has been incorporated in the model by
assuming that each bundle has three associated damping factors,
i.e. SCST> 8 E N D a n d SpAR' In a d d l t l ù n > provision has been
made to introduce extra structural damping forces at the end
constraints of the bundle string.
For convenience, it has been assumed that the extra generalized
damning forces at the constraints are proportional to the
endplate and central support tube stiffnesses at the ends of
the bundle string. e.g., the generalized endplate damping
forces associated with the first bundle will be
gENDl ( KEND1 + K END2 ) t ( ' l ~ X gENDl
iDETOPX (60)
where the last term is the extra damping associated with the
end constraint spring K . DETOPX is the 'extra' structural
damping factor operating on the endplate stiffness at the top
of the bundle string.
-60-
The additional damping forces at the constraints are formed
into two square matrices SDCSTX and SDENDX corresponding to
the endplate and central support tube damping respectively.
SDCSTX=
(i.DCT0PX.KCST1) 0
and
SDENDX=
(i.DCTOPX.KEND1) 0,
" • 0
0 .' -0 (i.DCBOTX.KCBOT)
• •. o
0 : : .0 (i.DEBOTX.KEBOT) (M89)
DCTOPX and DCBOTX are the extra structural damping factors
associated with the central support tube constraint springs
at the top and bottom of the fuel assembly. DEBOTX is the
extra damping factor associated with the bottom endplate
cons traint.
Matrices (M88) and (M89) are added to the potential energy
coefficient matrices V. .. and V^ respectively. This was
mentioned in section 2.5.2.
- 6 1 -
4.3.2 Comput atlon
The end const ra in t s t i f fness terms KCBOT and KEBOT are input
into subroutine PINNED, and are subsequently t ransferred to
the main program through statement DYNMOD 227. KCBOT and
KEBOT are then added to the coeff icient matrices V and V
respect ively to form the matrices given by expressions (M86)
and (M87); see statements DYNMOD 259 and 261.
The ext ra s t r u c t u r a l damping terms associated with the
cons t ra in t applied to the end of the l a s t bundle are assembled
into matrix form in statements PINNED 86 and 88. These
matrices are then t ransferred to the main program (DYNMOD 227)
and added to s imilar matrices describing the extra damping at
the upper cons t ra in t . Statements DYNMOD 2 70 to 2 75 form the
matrices (M88) and (M89). SDÇSTX and SDENDX are then incorpora-
ted in matrices ^ , and ^ in statements DYNMOD 279 and 2 82.
-62-
TRANSIENT RESPONSE TO HARMONIC FORCES
A fuel assembly will experience a transient vibration when
it is forced to change from one steady state of vibration to
another. Fluid cross-flow, acoustic phenomena, and differen-
tial pressures across the assembly are examples of forces
which may be responsible for such motion. If a bundle string
is initially at rest in its equilibrium position, the applica-
tion of constant, phase related, harmonic forces of the same
frequency will cause the structure to vibrate with increasing
amplitude. Eventually the string will settle down to a
steady state of vibration.
When the forcing frequency is very near to a natural frequency
of the fuel assembly in parallel flow, resonance may occur.
Equally, transient perturbations of the bundle string may
trigger fluid elastic instabilities. In these circumstances,
the fuel will vibrate with an increasing amplitude until it
contacts the wall of the flow channel. Fretting, rattling
and tapping wear processes will result causing damage to the
fuel and flow channel.
Under certain conditions, interactions between fuel and channel
will induce the former to lose energy to the channel and fall
back towards its equilibrium position. The harmonic forces
will continue to excite the fuel from its low energy state
and the non-linear cycle of events will be repeated.
This chapter describes the method used to predict the
transient response of a fuel assembly in axial flow to a
number of steady, phase related, harmonic forces of the same
frequency. The derivation of the matrix expressions used for
the transient response calculations has been renorted in detail.
This will assist in the subsequent development of DYNMOD to
include solutions of non-linear interaction problems where the
applied forces are not harmonic.
-6 3-
5.1 Transient Response Theory
The equations of motion of the bundle string when subjected
to the N generalized forces Q, are given by the matrix
expression
A. i + l i . $ + C . < \ > = Q (M90)
The left-hand side of (M90) is identical to (M3). Coefficient
matrices A, B and C were derived from the structural and fluid
kinetic energies, the gravitational and strain potential
energies, the generalized structural damping forces and the
generalized hydrodynamic forces.
Solutions of (M90) will provide absolute values of the bundle
angles <f>, i . e . not normalized eigenvectors.
5.1.1 Uncouple the Equations of Motion
As in section 2.9, equation (M90) is reduced to a f i rs t order
differential equation
T
BR . z + ER . z = F (M91)
Ôwhere 3R, ER and z were defined for equation (M65), and F = (-=r)
(M92)
For N degrees of freedom, BR and ER are of order (2Nx2N), and
column vectors z and F are of order (2Nxl).
In order to uncouple the equations in (M91), a knowledge of
the eigenvalues and eigenvectors of the unperturbed fuel/
fluid system is required. i.e., The homogeneous equations
given by (M3) must be solved. These solutions were discussed
in sect ion 2.9.
-64-
Matrix equation (M91) can be uncoupled by expressing the
generalized coordinates z(x,t) in terms of the modal matrix
V(x) and a set of normal coordinates Z(t),
z(x,t) = V(x) . 3(t) (M93)a.
(The modal contribution method for uncoupling these equations
of motion is valid because the fuel/fluid dynamic charac-
teristics are described uniquely by the modal matrix V(x)
and the eigenvalues y . )
Substitution of (M93) into (M91) gives
B R . V . 2 + E R . V . 3 = F
'Xi *\i '"KJ 'XI
Premultiplication by BR gives
V . 2 + B R " 1 . ER . V . J = B R - 1 . F•\j I j "u "u 'X,
Similarly, premultiplication by V leaves
3 + V " 1 . B R - 1 . ER . V . Z = V " 1 . B R - 1 . F (M94)
Expression (M94) can be simplified by considering the following
relationship between the eigenvalues and eigenvectors of
the homogeneous equations of motion.
If u is the diagonal matrix form of the eigenvalues p, then
all of the solutions of expression (M65) can be represented
-65-
by the matrix equation
Z(x,t) = V(x) . exp[i*y*t] (M95)
where Z(x,t) = (Z , Z Z'Xi 1 i J •V- Z is formed by
the horizontal joining of Z., j = 1 to 2N, the harmonic
solutions of equation (M65). The matrix is of order (2Nx2N)
and satisfies the expression
BR . Z + ER . Z == 0 (M96)
The diagonal matrix °xp[i*;i*t] in (M95) is constructed as
f o Hows :
e x p U u ^ t ) 0 0,
0 exp( ip 2 2 t ) 0
Different ia t ion of equation (M95) u'ith respect to time gives
Z(x,t) = V(x) . (i*u) . exp[i*y*t]1/ 'V 'V 'b
(M97)
','iiere (i*y) is the diagonal matrix
iu 22
2N2N
-66-
Substitution of equations (M95) and (M97) into equation (M96)
leaves
BR . V . (i*M) . exp[i*y*t] + ER . V . exp[i*y*t] = 0
Post multiplication by (exp |_ i*U*t ] ) gives
BR . V (i*p) + ER . V = 0r\j o. a. TJ a. %
Premultiplication by BR leaves
V . (i*u) + BR"1 . ER . V = 0
Finally, premultiplication by V gives the relation
(i*U) = -V"1 . BR 1 . ER . V (M98)a- r\j % ^ %
S u b s t i t u t i o n of (M98) in to (M94) provides the uncoupled
s e t of e q u a t i o n s ,
2 ~ (i*V>) • S = V~ . BR . F (M99)
in the normal coord ina tes 2 ( t ) .
-67-
5.1.2 Solve the Uncoupled Equations
Equation (M99) can be solved using the following identity
d— (exu[-i*u*t]) = -(i*p) . exp[-i*y*t]dt ^ ^ ^
= -exp[-i*p*t] . (i*y) (M100)
( i . e . , All matrices in (M100) are diagonal).
Pretnul t iplying (M99) by exp[-i*y*t] and post-multiplying (M100)
t'y 2 gives
exp[-i*u*t]. 2 = exp[-i*M*t] . (i*p) .2 + expE-i*y*tJ. V . BR~ . F"V r \ , *\j *\j % *\j
and d~ (expE-i*u*t]). S = - exp[-i*y*t] . (i*p) . 2
Addition of these two matr ix equa t ions leaves
- d ~ 1 1 ~pxp[-i*u*t]. 2 + — (expL-i*U*t]>."t= exp[-i*u*t] .V~ . BR~ . F
% dt ' ^
The terms on the l e f t - hand s ide of t h i s equa t ion are the
d e r i v a t i v e of a p roduc t . Therefore the uncoupled equa t ions of
mo t ion be come
d[exp[-i*y*t].2] = expL-i*H*t] . V~ . BR"1. F ( M 1 0 1 )
d t ^ % 'x, <\,
-6 8-
The normal coordinates 2 ( T ) of the fuel assembly, after time T,
are found by integrating (M101) from 0 to T. This leaves
exp[-i*u*T]. J(T) - S(o) = / expC-i*U*t] .v"1. BR"1 .F * dt
If the bundle string is in its equilibrium position at time
t = 0, then the initial reference coordinates S(o) can be
set equal to 5. Premultiplicat ion of (M101) by (exp[-i*y*t])~
gives
(T)= (exp[-i*u*t]) 1 . / exp[-i*u*t]. V 1. BR"1. F * dt (M102)
Because the inverse of a diagonal matrix is a diagonal matrix
of reciprocal elements, i.e.
exp(-iu T)
then equation (M102) becomes
T
:(T) = exp[i*p*î]. / exp[-i*p*t]. v"1. BR"1 .F * dt (M103)
-69 -
This matrix equation gives the normal coordinates of the
bundle s t r i n g r seconds a f t e r the forces F are appl ied .
For the case of harmonic applied forces of the form
F = P * expiut (M10A)
where the vector P contains amplitude and relative phase
information, and u is the angular frequency of the applied
forces, equation (M103) becomes
= exp[i*u*t]. / (exp[-i*u*t]*exp lot). v"1. BR"1. P * dt (M105)
o
Here, expicot is a scalar multiplier, therefore
(exp[-i*ii*t]*expiwt) = xexp(iw-ip . ) t 0
0 exp (iw-iu ) t
0,
After evaluating the integral, expression (M105) becomes
3(T) = S . V"1 . BR"1 . ,7 (M106)
where the diagonal matrix S is the time function
-70-
S =a.
io)T iu Te -e Jj
i(w-U_. J
J-1.2N
(e±a)T-eilJ2N2NT)
Equation (M106) i s transformed back into the generalized
coordinates of the bundle s t r ing by pre-multiplying by the
modal matrix V(x) .
Z(T) = V . S . V% <\, a.
(M107)
Because z = IT/, the upper half of the vector given by (M107)
consists of the bundle angular velocities and the lower half
consists of the angular deflections after time T.
Defining an operator I, such that (MAT) is the lower half of
the matrix or column vector MAT, then
H T ) = (V . s . v"1 . BR" 1 . T ) L (M108)
I f t h e l a t e r a l d e f l e c t i o n s a t t h e b u n d l e e n d s a r e y ( T ) , t h e n
f r o m ( M 6 9 ) ,
y ( x ) = CM AT . <|>(T) (M109)
where CMAT is a coordinate transform matrix defined by
- 7 1 -
2l2 * 3
There fore, the l a t e r a l def lect ion of the bundle s t r i n g T
seconds af te r appl ica t ion of the forces P*exp(i6Jt) is given
by
Ly(T) = CMAT. (V. S. V . BR . P) (MHO)
'V O. 'X, % %
The elements of the column vector y(T) are complex numbers.
Because P*expiwt = P* ( costot + isinwt) the real part of y(t) will
describe the response to a cosinusoidal force applied at t=0.
Similarly, the imaginary part of y(x) will describe the response
to a sinusoidal force applied at t = 0.
If the vector y(f) is evaluated at incremental times, then
its elements will represent the time domain transient response
of the bundle ends to the applied harmonic forces.
5.1.3 Generalized Harmonic Forces
The square matrix BR and modal matrix V in (MHO) have been
defined in section 2.9, and the time integral matrix S was
discussed in the previous section. P, the column vector des-
cribing the magnitudes and phases of the generalized applied
forces, will now be examined in a little more detail.
- 7 2 -
With reference to Figure 10, the column vector of forces
(f), acting in the y direction, cause displacements y.
The work done in producing these bundle deflections is
Similarly, the generalized forces Q acting on the bundle
string will cause angular deflections <}>. Hence the work-T
done by the generalized forces is to = Q • <J>. For the sameO
def lec t ion of the bundle s t r i n g in each case , U) and to can be
equated. This gives-T - -T -f . y = Q . <J) = t o t a l work done on the s t r i n g .
S u b s t i t u t i n g for y from equation (M109), the above expression
becomes
fT . CMAT . $ = QT . cj>
Therefore, by comparing both sides of the matrix equation
-T -Tf . CMAT = Q ;
the transpose of t h i s equation gives
Q = (CMAT)T . f (Mill)
-73-
If f is of the form p* exp(iwt), then from equations (M92)
and (M104),
Using this equation, the vector p which describes the ampli-
tudes and phases of forces applied to the ends of the bundles, can
be converted into the generalized force vector P. (Element j
of p is a complex number describing the amplitude and phase
of the force acting in the y direction at the end of bundle j.)
5.2 Computation
The time domain transient response of a fuel assembly to applied
harmonic forces is calculated by the subroutine FORCES.
Given the force vector FORCE ( = p) and the forcing frequency
OMEGA(=u/27r) , the subroutine will evaluate y(i) in equation
(MHO) at times T which vary from zero to TMAX in steps of
SAMPLE.
The generalized force vector FBIG (=P) is evaluated first in
statements FORCES 75 to 77. Column vector GENFOR (sy"1 . BR". 1?)
is calculated in FORCES 78. Construction of the diagonal
matrix INTIME (=S) takes place in statements FORCES 88 to 93.
Generalized coordinate matrix z, given by equation (M107), is
evaluated in FORCES 103.
A rectangular matrix of angular displacements corresponding to
different values of TOR (^T) is formulated in FORCES 104 to 109,
i.e. PHI = (<i>(T0) (KTj) ï(T2) cj)(TK)), where K= TMAX/SAMPLE.
Finally, the rectangular matrix of order (NxK) of bundle end
deflections DEFLEC is given by FORCES 110; this matrix is then
printed in statement FORCES 114.
-74-
6. STEADY STATE RESPONSE TO HARMONIC FORCES
If the harmonic forces discussed in the previous chapter are
applied to the fuel assembly, its vibration amplitudes will
eventually build up to steady maximum values. When the applied
forces are excessive, or have a frequency near a natural
frequency of the fuel/fluid system, mechanical damage may
occur because of the large response. Therefore, in order to
avoid fatigue failures, fretting wear and pressure seal damage,
a knowledge of the steady response of the fuel is desirable.
The limiting response of a fuel assembly to gradually applied
harmonic forces is calculated in subroutine FORCES.
6.1 Theory and Matrix Formulation
The uncoupled equations of motion of the bundle string in
parallel flow are given by expression (M99), i.e.
3 - (i*M) • X = V"1 . BR"1 . F (M99)
where Z(t) are the normal coordinates. The steady harmonic
solution of this equation will take the form
5 = S * exp(iwt) (M113)
where S is a vector of forced vibration amplitudes and relative
phases in the normal coordinates, and ui is the angular frequency
of the forces. Differentiating (M113) with respect to time
gives
<£ = Ë * (iw) * exp(ifjjt) = (iw) * H * exp(itot) (M114)
Substituting (M113) and (M11A) into (M99) leaves
(iw) * S * exp(iut) - (i*y) . jr * exp(icot) « V*1 . BR"1. F
-75-
Substituting for F, from equation (M10A), and dividing by the
scalar exp(iwt) , the above equation becomes
. - = v"1 (Ml15)
Because lu* 2 «f -{lui*I ) . H, equation (M115) can be rearranged. '.' . 'x,
to give . - ,"'••
- i*p) . Ê = v"1 . BR"1 . p (M116)
where the square matrix (iw*I - i*u) is defined as
Pre-mul tiplication of equation (M116) by (ito*I - i*U) -1
leaves
(iw*r - i . v"1. BR" 1(M117)
Using expression (M93), (M117) becomes
Z = V . (ia)*I - i*M)"1 . V"1. BR"1. p (M118)
where Z are the amplitudes and phases of the steady harmonic
response in generalized coordinates. From the matrix expression
-76-
, the steady state bundle angular deflections are
(to) = (V . (io)*I - i*y)~1 . V"1 . BR"1 . p ) L
Therefore the steady amplitudes and relative phases of the
bundle end deflections are, from expression (M109),
Y(w) = CMAT. (V . (iw*l - i*^)" 1 . V"1 . BIT1 . p ) L
If the phases of the applied forces are measured relative to
reference force, then the phases in the calculated response
are also taken relative to the reference force.
The amplitude and phase of the steady response at the end of
bundle p is given by the p element in the vector Y. This
corresponds to a single spectral line at the frequency U/2TT.
Whereas the transient response results of expression (MHO)
are in time domain, the steady harmonic response results
of (M119) are in frequency domain.
6.2 Computation
a
f TT " T> n «» The column vector GENFOR (=V .BR .P) is evaluated in statement
FORCES 78. Diagonal matrix FRECK ( = (ia)*I - i*y)) is calculated
in FORCES 130. The amplitude and relative phase of the steady
response given by ZED = V.FRECK~ . GENFOR is evaluated in FORCES
132. Bundle angular response PHI, which is equivalent to the lower
half of ZED, is calculated in FORCES 133. Finally, the complex
number vector P0LAR(s Y(w)), giving the amplitudes and relative
phases of the steady state harmonic response, is calculated in
FORCES 135. FORCES 139 prints the response vector POLAR.
-7 7-
7. APPLIED FORCES IN A CONSTRAINED SYSTEM
7.1 Theory and Matrix Formulation
In section 4.2 a method of reducing the order of matrices A,
B and C, in the homogeneous equation of motion (M3), was
described. Here, the generalized coordinate reduction
matrix R was used to reduce the number of degrees of freedoma.
of the i d e a l i z e d fue l assembly. A s i m i l a r t echnique can be
a p p l i e d to the inhomogeneous e q u a t i o n
A . 3> + B. <j)+C.<J> = Q (M90)
From equa t ion (Mi l l ) t h i s becomes
A.c |> + B.<J) + C.<!> = (CMAT)T.f
Using (M73) to s u b s t i t u t e for the (f> c o o r d i n a t e s ,
A.R.tf' + B .R .0 ' + C.R.ci>' = (CMAT)T . f
TF i n a l l y , by p r e - m u l t i p l y i n g by R , the reduced equa t ion of
motion i s o b t a i n e d ,
RT.C.R.<f)' = RT.(CMAT)T . f (M120)
T T —
The term R .(CMAT) . f in (M120) i s the r e d u c e d g e n e r a l i z e d
f o r c e v e c t o r of o r d e r (N-M)xl , where M i s t h e number of
c o n s t r a i n t s . Column vector f con ta ins the N forces a c t i n g
in the y d i r e c t i o n a t the bundle ends . After c a l c u l a t i n g
the s o l u t i o n s <j> ' of equat ion (M120), the bundle angles 0
can be eva lua ted from express ion (M73).
-78-
7.2 Computation
The applied forces (FORCE, in rectangular Gartesian coordinates)
are generalized in statement FORCES 75. If the fuel
assembly is constrained, the force vector is reduced in
FORCES 76. The generalized force vector (F) is then
used in the calculation of transient and steady response.
After evaluation of the response data in reduced form, the
absolute bundle angle deflections are calculated in statements
FORCES 109 and 134. Knowing the column vectors PÏÏI, the
absolute bundle end-deflections DISPLA (for transient response)
and the steady amplitudes and phases POLAR can be calculated;
see FORCES 110 and 134.
-79-
8. TRANSFER FUNCTIONS
From Chapter 6 it can be seen that a sinele lateral
force, of frequency W/2TT and amplitude and phase given by p
applied at the end of bundle p, will produce a displacement
response along the bundle string represented by Y . The
elements of
are complex numbers describing the amplitude and phase of the
response at the bundle ends. For forces applied at different
bundle ends there will be correspondingly different column
vectors of response.
In a steady dynamic system, where the mechanical and fluid
parameters are not varying, division of the response vector
Y by the force p will produce a column vector of constantP _Pcomolex numbers h . The element h. of h describes the dis-
P JP P
placement and relative phase angle (between displacement and
force) at the end of bundle j caused by unit force acting at
the end of bundle p.
For a fuel string consisting of N bundles there will be N
column vectors h . Therefore, the force-deflection propertiesP
of the dynamic system at frequency o)/2ir can be completely
described by a square matrix H(w) given by the expression
(p)D
- 8 0 -
wbere Y = (Y, Y,, Y,% 1 2 3
N2
' • y2N
(M122)
Hh12 •••• hlN
h21 h22 •••• h2N
V hN2 •••• hNN(M123)
and (p)
(M124)
The transfer function matrix H(w) is a constant and i s a uniquea.
property of the dynamic system. H(u) has the dimensions.of
displacement per unit force and is called the receptance (or
compliance) of the system. The matrix denoted by H(œ) (where
H((JÛ) = iu*H(u)) has the dimensions velocity per unit force and
is called the mobility of the system.
-81-
Elements h and h , where r^s, are known as cross-receptancesr s r s
and cross-mobilities respectively. Elements with r=s are
called auto, or driven point, receptances and mobilities.
A knowledge of these transfer functions over a range of
frequencies will enable the engineer to predict the response
of a fuel assembly to applied forces in this spectral range.
The receptance and mobility of the fuel assembly are functions
of the properties of the dynamic system. Therefore, computer
experiments in which the structural and fluid parameters
(eg. hydrodynamic mass, viscous damping, stiffness, geometrical
shape and streamlining) are varied can help in the design of
fuel which meets the required transfer function specifications.
The subroutine RECMOB calculates the column vectors of
receptance h and the column vectors of mobility h . TheP P
complete receptance matrix H(w) is calculated in subroutinesRANDOM and CALFOR.
8.1 Theory and Matrix Formulation
The receptance and mobility vectors h and h of the fuel
assembly are equal to the displacement and velocity vectors
resulting from a pseudo unit force acting at the end of bundle
p. Using the method outlined in section 6.1 for steady harmonic
response, the displacement and velocity vectors are given by
Y (u>) = h (w) = CMAT.(V.(iu)*I - i*;!)""1 . v"1 . BR"1 . P ) L (M125)P P ^ <\, <\, <\, -v, 'v, u
and
iw*Y (w) = h (u)) = CMAT.(V.(iu*I - i*^)"1 . v"1 . BR"1 . P ) U (M126)P P % % % "u <\ i\, U
-82-
(In expression (M126) the operator (...) denotes the upper half of the
matrix or column vector in parenthesis.) The vector P is the generalized
force corresponding to a unit lateral force acting at a point on the
bundle string, i.e.,
P = I _ | (Ml 27)' (CMAT) . u
where
If the p element of vector u is set eaual to unity, then
auto- and cross-receptances and mobilities between the end ot
bundle p and the ends of the other bundles in the string are
calculated.
In RECMOB the diagonal matrix (iw*I - i*vO is calculated
for an incrementally increasing frequency. Substitution in
(M125) and (M126) then gives the transfer functions over a
specified range of frequencies (see also section 9.1).
8.2 Computation
For the calculation of the transfer function vectors h (w) and- Ph (to), the number of the bundle, at the end o^ which theP
pseudo unit force (or shaker) acts, is specified.
-83-
The column vector UNIFOR (= u, in expression (M127)) is
constructed in statements RECMOB 64 to 68. UNFB~IG (=V..) ,
UNÏÏËNF (=V~1.BR~1.P- ) and CHANGE (= io)*I - i*vi) are
calculated in statements RECMOB 69 to 76. Hence, h (eu) andP
h (a)) are evaluated in RECMOB 78 to 84. i.e., after
ZEDNEW (= V. (iu*I - i*»)"1 . V"1. BR"1. P ) is calculatedAi % l\i <\j r\, u
i n RECMOB 7 8 , PHIMOB (= $ = (ZEDNEW)U) and PHÎREC (s<j> = (ZEDNEW)L)a r e s e p a r a t e d i n RECMOB 79 t o 8 2 . The v e c t o r s MOBIL(S h )
p
and RECEP (= h ) are obtained by transforming PHIMOB and PHIFEC
from the generalized coordinates; see RECMOB 83 and 84.
Because of the difficulty in directly interpreting the raw
data in MOBIL and RECEP, each complex element is converted into
polar form giving amplitude and relative phase angle informa-
tion. These conversions take place in statements RECMOB 88 to
100.
The amplitude of an element h. is given by
|h3P
_ 1r e l a t i ve phase angle i s 6 = tan (h. -./h. ) .
j P 1P J P
Here h. is the real part of h and h. is the
imaginary part. The amplitude vector of receptance is
MODREC (=|h |) and the vector of receptance phase angles is
PHASER ( = 8, ). The amplitude vector of mobility is MODMOB.7 hp "
(= h I) and the vector of mobility phase angles is PHASEMP
(=9- ). These four vectors are evaluated over a range of
frequencies, and are printed at each frequency step by state-
ment RECMOB 107.
-84-
9. RANDOM EXCITATION
The forces, fluid or otherwise, acting upon nuclear fuel
assemblies are generally of a complicated nature. Very
often it is difficult to predict how the forces will vary
with time. It is therefore necessary to resort to a statis-
tical description of the forces for forced response calcula-
tions .
Bundle string response can be deduced from a knowledge of
the transfer function characteristics of the string and the
statistics of the applied forces. Subroutine RANDOM is
designed to calculate the response of a fuel assembly in
axial flow to such complicated applied forces.
RANDOM will predict the response to applied forces when they
can be described as
(i) discrete force spectra,
(ii) averaged power spectral densities, or
(iii) averaged auto power spectral densities, for the case
of uncorrelated forces.
For a discrete force spectra input, the output consists of
the complex response spectrum (metres), the auto power2 2
(intensity) spectrum (metres ), the mean square value (metres )
and root mean square value (metres) of response at each bundle
end.
If power spectral densities of force are input, the output2
consists of the auto power spectral density of response (metres /
Hz), the mean square value and the root mean square value of
response at each bundle end. With minor adjustments to the
subroutine, the cross power spectra of response will be
printed. The basic theory used in subroutine RANDOM is discussed
be^ow.
-85-
9.1
9.1.1
Construction of the Square Receptance Matrix H(o))'or
Theory and Matrix Formulation
For a fuel assembly of N bundles, the (NxN) matrix H(OJ)
completely, describes the receptance characteristics of the
idealized structure. H(oi) is defined by expression (M121).
It can be seen from expression (M123) that
H(u>) = (h 1 h 2 h 3 . . .SN) CMAT. - i*y)~1.V~1.BR~1.P_)L
(M128)
The square matrix P , of order (2NxN), is given by
(CMAT) . I
where the (NxN) identity matrix I is formed as a result of
horizontally joining the N unitary pseudo-force vectors,
i . e . ,
unit forceat
bundle 1
unit force unit forceat at
bundle 2 bundle N.
As (CMAT) . I = (CMAT)' (M129)
-86-
9.1.2 Computation
The generalized force matrix MATFOR (= P ) , given by equation
(M129), is evaluated in statement RANDOM 91.
Transfer function matrix TRAMAT (= H(w)) is then constructed
in statements RANDOM 91 to 100. The calculation of TRAMAT,
for a particular frequency, is similar to the evaluation of
the receptance vector described in section 8.2.
For N bundles, the order of TRAMAT is (NxN). If the end of
bundle p is pinned, then the p11*1 row and column of TRAMAT
contain zeros. Similarly, if the end of the bundle string
is constrained, the N row and column of TRAMAT will contain
zeros.
9.2 The Response to Discrete Force Spectra
Here the forces acting on the bundle string are assumed to
take the form shown in Figure 11. The forces exist only at
discrete frequencies within the frequency range of interest.
In the dynamic model the minimum forcing frequency (FMIN)
and spacing between adjacent spectral lines (BANWID) are
specified. The force matrix j) is of order (Nxm) . Each
column of p contains complex numbers which represent the
amplitude and phase of the forces acting on the bundle string
at a particular frequency.
-87-
9.2.1 Complex Response Spectrum
The amplitude and phase of the response at any frequency is
calculated using the expression below.
= H(u>) . p(w) (M130)
where p(co) is a column of p. This calculation is performed
for each of the m spectral lines; H(u>) is calculated at each
frequency.
The column vectors Y(u>) are horizontally joined, in spectral
order, to give the complete response spectrum matrix Y. Y is
of order (Nxm) and is the response equivalent to the force
matrix £. The rows of Y describe the amplitude and phase of
the response at the ends of the bundles. These are printed
in subroutine RANDOM.
9.2.2 Power Spectrum of Responsery
The discrete power spectrum (intensity spectrum) G'(co) (metres )R
of a single sinusoidal response is equal to its mean squarevalue (i.e., The power spectral density of response G (u>)
2(metres /Hz) goes to infinity at the frequency of the discrete
response). However, the integral of the p.s.d. over the
response frequency gives the mean square value or the 'power'
in the response at that frequency, i.e.,
,(w) . du (61)
-88-
The discrete auto power spectrum of response of the fuel
assembly is obtained as the (Nxm) matrix G', where
G 1 = 0.5 * Y * Y* (Ml 31)K
Y is the complex conjugate of the response spectrum matrix Y.% 'Xi
The elements of G' take the form 1/2 y. y* . They are meanr* J V J P
square values and therefore contain no relative phase
information. Each row of G' wi l l describe the discrete auto
power spectrum of response^at the end of a particular bundle.The rows of Gl are printed in subroutine RANDOM.
R
Mean Square and Root Mean Square Values of Response
The mean square value of the forced response at the end of a
bundle is given by the addition of the mean square values of
response at each discrete frequency,
i* v ? ; (MI32)
Element ¥„_ i s the total mean square value of response at end of bundle p.
The vector Y. contains the amplitude and phase of the response
of the fuel assembly at the j discrete frequency.
The root mean square value of the response i s given by the2
positive square root of 41 , i . e .
= |(f jg) I (M133)
2Column vectors ¥„_ and ^-wo are printed by subroutine RANDOM.
rib Krlo
-89-
9.2.4 Computation
The (Nxm) matrix DFORCE (=jj) of the force vectors p(to) is
input in statement RANDOM 116.
For a given frequency, the response vector DISVEC (=Y(w))
is calculated in RANDOM 118 using equation (M130). The
response vectors are evaluated for all the spectral lines
of interest and are joined horizontally in RANDOM 124 to
form the (Nxm) response matrix DISMAT (SY). The rows of Y,
which are the discrete response spectra at each bundle end
are then printed: RANDOM 134 to 139.
Statement RANDOM 144 evaluates the complex coniugate of the_ *
response matrix DMSTAR (=Y ). Hence the auto oower spectrum
of response DPOWER ?=G'), given by equation (M131), is
evaluated in RANDOM 145. The rows of DPOWER,
the autopower (or intensity) spectra of response at the bundle
ends, are printed in statements RANDOM 149 to 154.
,2Mean square displacements MSDISP (= V ) are calculated in
RANDOM 159 to 162 using equation (M132). The column vector
MSDISP is printed in RANDOM 163.
Root mean square displacements, RMSRES ( = * D M C ) , at the bundle
ends are evaluated in RANDOM 168 using equation (M133).
RMSRES is printed in RANDOM 169.
-90-
9. 3 Response to Forces Expressed In Averaged Power Spectral
Density Format
The forces acting on a fuel assembly in parallel flow will
rarely take the form of the discrete spectra described in
section 9.2. Very often, the forces are of a random nature.
Using special equipment, it is possible to obtain time records
of the forces acting on the components of a fuel assembly.
The frequency content of the force signals, measured at various
points along the string, will usually change from one time
domain record to the next (i.e., the amplitudes and relative
phases of the spectral lines vary with time). It is therefore
difficult to describe the forces acting on the fuel assembly.
The method generally adopted for the statistical representa-
tion of these forces is to average the power spectral density
matrix over many time records for each frequency point. This
matrix defines the average effective intensities and relative
phases of the forces acting on the bundle string. A probabi-
lity density function analysis of the time records will indicate
the distribution in amplitudes of the forces.
The theory used in subroutine RANDOM for the calculation of
fuel response from power spectral density of force data is
now discussed.
9.1.1 Power Spectral Density of Response
a) Two arbitrary random displacements u(t) and \>(t) can be added
to give a third random displacement 0)(t).
w(t) = u(t) + v(t) (62)
-91-
The autocorrelation function of this new motion is given by
R (T) = <u)(t) . w(t+T)>dill)
= <(u(t) +V(t)) . (u(t+T) + V(t+T))>
=<u(t) . u(t+T) + u(t) . V(t+T)
+ u(t+T) . V(t) + V(t) . V(t+T)>
• R « u ( T ) + Ruv ( T ) + Rvu ( T ) + Rvv ( T )'
where R (T) is the cross-correlation function between dis-
placements u(t) and v(t).
From the Wiener-Khinchin relation, the auto-power spectral
density function of displacement 0)(t) is proportional to the
Fourier transform of the auto-correlation function R ( T ) , i.e.,
+00
G (u>) = 2 f R (T) . exp (-io)T) . dT0)0) / U)U> yI
r2_oo
2/ <RUU<
T> + RU V
( T ) + R v u( T ) + Rvv ( T ) ) •
This equation then gives
• Guu ( w ) + Guv ( w ) + Gvu(a)) + Gvv(a)) <63>
G (a>) and G (ID) are cross-power spectral density functions.
-92-
Equation (63), which can be expanded to apply to the addition
of any number of random displacements, forms the basis of
the response calculations of this section.
b) The averaged power spectral density matrix of the forces
acting on the bundle string is defined here as
u T *G (at) = — * p(u) * [(p (ai)) * (ONES)] (M134)'v Aa> ^
where ONES is a (NxN) matrix of ones for a fuel assembly con-
sisting of N bundles.
Matrix G (to) describes the intensities and correlations of
the forces within the frequency band Au /2ir centred on the
angular frequency w.
Vector p(oj) consists of the 'equivalent' steady forces,
acting at the bundle ends, that will uniquely describe the
time averaged function G.p(w) at the frequency w/2ir. The
elements of p(u)) contain amplitude and phase information.
As an example, evaluation of (M134) for N=3 gives
_ * . ^ P1 P2 P1 P3
AuiP2P2 P2P3
P3P1 P3 P2 P3 P3
The auto-power spectral density terms on the diagonal are
force intensities; the cross terms elsewhere contain relative
phase and intensity information.
-93-
c) The matrix methods used by DYNMOD for the calculation of
random forced response are best described by considering the
three bundle string of figure 12.
Three 'equivalent' forces p., p and p , applied to the
ends of each bundle, will produce the complex displacements
y. , y and y,. The displacements and forces are related by
expression (M130).
Y (a.) = / y l \ = / H H H12 H13"
H21 H22 H23
H31 H32 H33'
where matrix multiplication gives
H12P2
H 2 2P 2
H32P2
Now, the vector of auto-power spectral densities of response
along the assembly, at the angular frequency 0), is defined
as
* - -*Gc = * Y * Y (Ml36)K Aw
Expansion of (M136) gives
(M137)
-94-
Substituting for y , y. and y. in equation (M137) from
(M135), the spectral densities become
* * * *
/ — - - Hl3P3 *lP*+ Yi— * [ (H2lPl + H22p2 + H23p3) (H21p* f H22p* + H*3p3)
(M138)
Consider the first equation in (M138); the auto-power spectral
density at the end of bundle 1 is given by
G R 1 1
•n
AGO
*( H U H U
*+ H11H12
*
*+ H12H13
P 1 P 1 •
P 1 P 2 -
*
P 2 P 3 "
" H12H12
H H12H11
" H13H11
h H13H12
P 2 P 2
*P 2 P 1
*P3P^
P 3 P 2 ( 6 4 )
The first three terms in (64) are each the auto power spec-
tral densities of response observed if the forces p. , p
and p_ acted on the bundle string independently. The
remaining terms are the contributions to the response caused
by correlations between the three forces.
Because the first three terms are phase independent, and
the other terms form complex conjugate pairs, the function
contains no relative phase information.
-95-
Expansion of expression (M134) gives
GF11 GF12 GF13
Gv>7F22
JT ^ / P1P1 • P1P2 P1P3 \A I P.P. n.p. n.p* 1
GF31 GF32 GF33/ \ p.p? p ^ * p ^ *
(Ml39)
From (M139), equa t ion (64) becomes
* * *GR11 = ( H11H11GF11 + H12H12GF22 + H13H13GF33
+ H11H12GF12 + H12H11R
* *+ H11H13GF13 + H13H11
+ H i2H13GF23 + H13H12GF32 ) (65)
Similarly, expansion of (M138) will give G D O. and G_-_.A Z £. KjJ
In general, for a fuel assembly consisting of N bundles,
the auto power spectral density of response at the end of
bundle p will be given by the equation
N N
G,, (OJ) = Y"* Y"* H 4 H * G,,. . (66)RPP / 2f P 1 PI F l 3
where H ., H . and G_.. are all functions of the angularpi PJ frij
frequency u).
-96-
The matrix equation for tha power spectral density of response
is
G_(ui) = H . G . (H*) T (M140)R F
where the (NxN) matrices G , G_ and H are a l l functions of co.
The auto-power s p e c t r a l densi ty of response i s given by the
diagonal of GD(u).R
i . e . GD(u)) = DIAG (G_(w)) (M1A1)
2In the dynamic model, i f G (w) has the un i t s Newtons /Hz, then
G (to) i s in metres /Hz.R
Subroutine RANDOM calculates the receptance matrix H(uj)
as described in section 9 .1 . The (NxN) power spectral density
of force matrix G_(a)) is supplied as input data for each
of m incremental frequencies. An example of the construc-
tion of G (OJ) for a two-bundle string and eight frequency
points is given in Figure 13.
9.3.2 Mean Square and Root Mean Square Values of Response
If the variable G (w) represents the auto-nower spectralRpp
density of response for the end of bundle p, then the mean
square displacement over the angular frequency range oii to 0)2
is given by
• /w l
i M e , G (tù) . dOJMSo / Rpp
-97-
Numerical integration of G (w) is achieved in RANDOMRpp
by adding the diagonals of matrix G_(CJ) for all of the m
frequency points considered. The total is then multiplied
by ACO/2TT the spacing between adjacent frequency points.
( M 1 4 2 )
_2The positive square root of ¥ gives the column vector of
MSroot mean square values of the response at the bundle ends.
9.3.3 Computation
The (NxN) matrix of the power spectral density o.e the forces
PSD£OR (SG (to)) is input in statement RANDOM 123 for each
frequency point.
RANDOM 187 forms the matrix TSTART (= (H*)T). Equations (M140)
and (M141) are then used to evaluate the diagonal of the
power spectral density of respi
frequency <JJ/2TT in RANDOM 188.
power spectral density of response AUTOPY (= G_(w)) at theK
Horizontal joining of vectors G (w) for all frequencies gives
the auto power spectral density of displacement matrix DISPOW.
The rows of DISPOW give the auto-power spectral density of
response at the bundle ends. These are printed in RANDOM 222
to 228.
Statements RANDOM 233 to 237 calculate the mean square_2
displacements MSDISP (5fMS) at= the bundle ends using equation
(M1A2). The vector MSDISP is printed in RANDOM 238.
-98-
Root mean square displacement vector RMSRES (= ? e ) isRMS
evaluated and printed in RANDOM 243 and 244.
9•4 Response to Uncorrelated Forces
9.4.1 Theory and Matrix Formulation
If the forces applied to the ends of the bundles of the fuel
assembly are uncorrelated then, on average, all cross-power
terms in the spectral density of force matrix vanish,
i.e., the elements G •+ 0 for many averages, where r ^ s.r rs
Hence, equation (65), in the three bundle string example,
becomes
GR11 " H11H11 GF11 + H12H12 GF22 + «13*13 GF33 ( 6 7 )
In general, the auto.power spectral density of response for
the end of bundle p is
N
\i HPJ GFJJ ( 6 8 )
where H . and G . . are. functions of the angular frequency a).
It can ba seen from equation (68) that the auto-power spectral
density of the response to uncorrelated forces is the sum
of the auto_power spectral densities of the responses to each
force acting separately on the fuel string. If the forces
are correlated, the auto-power spectral density of response
can be either greater or less than the value given by equation
(68).
-99-
Equation (68) can be expressed in matrix, form as
where H and G_, are functions of co.<\, Fu
GD is a vector of auto-power spectral density of responseRuto the uncorrelated autopower spectral density of force
vector G_ .Fu
Uncorrelated forces acting on a bundle string will give rise
to correlated displacements. Cross terms in the power spectral
density matrix of response are not calculated by equation
(M143).
The complete response matrix is given by equation (M140) by
setting the cross terms in the force matrix to zero. This
gives the equation
• I- ( 5 P U ) D • <Z )T (M144)
In subroutine RANDOM, G (<»)) is supplied as a (Nxm) matrix
consisting of N uncorrelated forces for each of m spectral
lines.
After calculating the receptance matrix H at a particular_ %
frequency, and knowing G at that frequency, the auto-powerr u
spectrum o .e response is calculated using equation (M143).2 °
If G_ is in Newton /Hz, then G_ will be in metres'/Hz.r u Ku
An example of the matrix of horizontally joined vectors G (10)Fu
is shown in Figure 14.
-100-
The mean square and root mean square values of the dis-
placement are found from the integrals of the auto-power
spectral density of response curves calculated for each
bundle end. The methods used to
were described in section 9.3.2.
~2bundle end. The methods used to evaluate ¥„_ and ¥_„„
MS RMS
9.4.2 Computation
A single (Nxm) matrix UNCORF of horizontally joined uncor-
related force vectors G (to), one for each frequency, are
input in statement RANDOM 205.
The matrix TSTAR (= H ) is calculated in RANDOM 210 and,
using equation (M143) , AUÏÔPY (= G_ ((*>)) is calculated inKU
RANDOM 211. The vectors of auto-power spectral density ofresponse, at a given frequency, AUTOPY are joined horizontally
to give the (Nxm) matrix DISPOW. This is undertaken in
statements RANDOM 213 to 218.
The auto-power spectral density of response at the bundle
ends is given by the rows of DISPOW. These are printed in
RANDOM 222 to 228.
Calculations of the mean square and root mean square dis-
placements from DISPOW are carried out in statements RANDOM'XJ
233 and 243.
-101-
10. RESOLVED FORCE CALCULATIONS
The vibration amplitudes of fuel assemblies in parallel
flow can be minimized by adjusting the structural and
hydrodynamic characteristics of the fuel/channel system as
predicted by the computer model.
However, in addition to the hydrodynamic forces considered
in the formulation of the model, vibration energy can be
transferred to the fuel assembly by other mechanisms. These
forces can cause the assembly and its component parts to
vibrate with amplitudes in excess of the design specifications,
(Such forces might include cross-flow excitation, acoustic
excitation, flow pulsations, unbalanced pressure forces
caused by flow turbulence, and mechanical vibration forces
transmitted through the fuel assembly supports or the fluid.)
For research and design purposes it is often desirable to
know the nature of the lateral forces acting on a prototype
fuel assembly. If the complete response can be measured,
and the transfer functions of the fuel/fluid system are known
(by experiment or calculation), then the characteristics of
the resolved forces can be deduced using subroutine CALFOR.
Once the forces and points of action are known, their impor-
tance can be assessed. The prototype design can then be
altered to reduce the intensity of the forces or minimize
their effect.
The theory used in the force calculation subroutine CALFOR
is now discussed.
-102-
10.1 Calculation of Discrete Force Spectra
10.1.1 Theory and Matrix formulation
If the response measured along the fuel assembly can be
described in discrete spectra form, then the spectra of
the applied forces will also be discrete.
The amplitudes and phases of the forces are related to those
of the displacements by the receptance matrix. This is
seen from equation (M130), i.e.
Y (to) = H((0). p(0)) (M130)
If both sides of this equation are pre-mul t ip l i ed by H (to) ,
the force vector i s defined by the response vector
p(cj) = H~1((JD) . Y(w) (M145)a,
Evaluation of this equation for a particular spectral line
will give the amplitudes and phases of the applied forces
resolved at the displacement measurement stations.
In subroutine CALFOR the displacement vector Y(w) describes
the harmonic response at the ends of each of the N bundles
of the fuel assembly. Therefore p(u) will be a (Nxl) vector
of resolved forces at the bundle ends.
If p(<rt) is evaluated for each of m spectral lines, and the
vectors are joined horizontally in spectral order, the
discrete force spectrum matrix j is formed. The j row of p
will give the discrete force spectra at the end of bundle j.
-103-
The cross-power (intensity) matrix of the resolved forces,
at the angular frequency w, is given by the expression
GI(.O}) -= 0.5 * (p(u) * (ONES)) . (p*(a)))D (M146)
Auto-oower ( intensi ty) spectra of the resolved forces are
obtained from the expression
G*' = 0.5 * p * p* (M147)
I'The elements of G contain no phase information.
— 2Evaluation of the mean square force vector *F_W_ is achieved
(Mb
using the expression
m
2 Î (H148)
r KMb
where p. is the force vector at the j discrete spectral line.
The root mean square force vector "F is given by the positive
p" r KMbsquare root of V , i.e.
(M149)
In DYNMOD, for a response Y(uo) measured in metres, p(w) will1 2 2
be in Newtons and G (u) will be in Newtons (not Newton /Hz)
-104-
10.1.2 Computation
In CALFOR, the receptance matrix is calculated using equation
(M128) in statements CALFOR 96 to 110. Provision exists,
however, for an experimentally determined receptance matrix
to be input directly into the subroutine; see CALFOR 101.
The (Nxm) matrix DISMAT (=Y) consisting of horizontally
joined displacement vectors for each of the m discrete spec-
tral lines, is input in statement CALFOR 125. CALFOR 129
calculates the force vector FORVEC (sp(u))) for each spectral
line using equation (M145).
The cross-power spectrum of force MPSFOR (=G (w)) isa, %F
calculated for each spectral line in CALFOR 135 using
equation (M146). MPSFOR is printed in CALFOR 136 to 145.
Force vectors FORVEC, for the m spectral lines are horizon-
tally joined in order of increasing frequency to give the
discrete force matrix DFORCE (sp ) in CALFOR 157 and 162.
The discrete force spectra acting at the bundle ends are then
nrinted in CALFOR 166 to 171.
The matrix FPQWER (=£F ) describing the auto-pOwer spectrum
of force given by equation (M147) is evaluated in CALFOR 176
and 177. FPOWER is printed in CALFOR 181 to 186.
— 2Mean square values of resolved force MSFOR (=V „) are
FMS
evaluated via equation (M148) in CALFOR 191 to 194. MSfFOR
is printed in CALFOR 195.
The root mean square values of the resolved forces RMSFOR
(B^D«c) a r e calculated in CALFOR 200 and printed in CALFOR 201,
-105-
10.2 Resolved Forces Calculated from Power Spectral Density of
Response Data
10.2.1 Theory and Matrix Formulation
The cross-power spectral density matrix of force G_(u)) , at
the angular frequency (i), can be deduced from equation (M140)
G_ = H . G . (H*)T (M140)<v,R % %? i/
where G and H are also functions of w.<\,R %
TPostmultiplying (M140) by the inverse of (H*) gives
Finally, pre-multiplication of the new expression by H
leaves
G_ = H"1 . G_ . [(H*)1]"1 (M150)
Hence G can be calculated from the cross-power spectral
density* of response and the receptance matrix.
In subroutine CALFOR, a (NxN) matrix of complex numbers £
is input for each of m frequency points. The corresponding m
matrices of G_ are calculated and printed.
Auto-power spectral density vectors of the resolved forces
G_ are given by the diagonals of the matrices G (w).
G_(u>) = DIAG (G (a))) (M151)
-106-
A p l o t of the p element of vec tor G (ui) aga in s t w w i l l
de sc r ibe the auto-power s p e c t r a l dens i ty of the forces
a c t i n g a t the end of bundle p . I n t e g r a t i o n of t h i s curve
w i l l give the mean square of the forces: artinf* at 'lie end
of bundle p . In vector form, the mean squares o •" Un-
resolved forces a c t i n g on the s t r i n g are -jivnn by
( m »
S •'•)( M 1 5 2 )
where Aw i s the i n t e r v a l hefween H-tjactTt frpquonrv r-.inf.s
and m i s the t o t a l number of point '- in ftoqurrirv l 'cr . i in.
Corresponding root mean square values arc given by the p o s i t i v e
square root of the mean square v a l u e s .
10 .2 .2 Computation
For each frequency, a (NxN) r r o s s "own <;pi av r i l t'eus] !:y
matr ix of d i sp lacements DISPOW (s G_(w)) i s input i n to
subrou t ine CALFOR in s ta tement 213.
The cross-power s p e c t r a l dens i ty matr ix of force PSDFORa.
(=G_(w)) is then calculated in CALFOR 219 and 220 using
equation (M150). PSDFOR is printed, for each frequency, in
statements CALFOR 222 to 229.
AUTOPF, the auto-power spectra] drnsitv vector nf i"orev> is
calculated, according to equation (Ml 51), in CALVOR 212.
The vectors AUTOPF, calculated for each frequency, are joined
horizontally in spectral order to form the matrix FORPOW
in statements CALFOR 2.38 and 2A3. The rows of FORPOW,
giving the power spectral densities of the forces acting at
the bundle ends over the frequency range of interest, are
printed in CALFOR 247 to 253.
-1Û7-
The mean square values of for_c_e at the bundle ends are2
I'iven bv the vector MSFOR (S*1'FMS in equation (M152)).
MS FOR is evaluated in CALFOR 258 to 262 and pr in ted in
"MFOR 263, The corresponding root mean square values of
••be forr-es RMSFOR are ca lcula ted in CALFOR 268 and pr in ted
;n CALFOR 269.
10 . 3 _ïhe_ Inverted Receptance Matrix
10.1- .1 Thi?ory and Matrix Formulation
Ts>; recentance matrix H(ai) of order (NxN) for a s t r i n g of
-•' bundles i s given by equation (M128). If the idea l i zed
•. ue 1 assembly i s constrained <; r supported at one po in t ,
..o in Figure 1, invers ion of H(iu) usually presents no problem.
However, if the bundle s t r i n g is constrained at other points
si.ina i t s length , as shown in Figures 7, 8 and 9, then H(w)
\.s s i n g u l a r , i . e , i t s inverse cannot be defined.
The s i n g u l a r i t y of H(o)ï a r i s e s because i t s p row andr. •,'iiran w i i 1 contain zeros if the end of bundle n i s constrained.
i'.rynai Ae ". the following example-..
Acccr Hng to equation (M121), the response matrix of the
:.'.ri:i;! of six bundles shown in Figure 15 i s r e l a t ed to the
.": i. i •'-,-•.• - i . i i f o r : e TOrîtrix a s f o l l o w s !
• | •'}.?
Y Cw)
13
,D
y l 4
-V24y34«7
'V
Y15y 25y35y 45y55V65
y16~y 26•yr
3D
y 46V56Vn6
" H l l
H21H31H4JH 51
_ H 61
H12H 9 ?
H32H42H52H62
«13H23H33K/<3H53H h3
«14
«24H34
"/,/,
H-,
H
H
H
H
15
25
35
AS
S 5
; 5
H 16~
«26H3r,
« 6 h
n l0
0
0
0
0
0
P 20
0
0
0
0
0
P30
0
0
0
0
0
P 40
0
0
0
0
0
p 50
0
0
0
0
0
p
(M153)
-108-
Here the column j of Y(w) describes the response of the
bundle string to the single force p. acting at the end of
bundle j. The six forces do not act together.
If, for example, the ends cf the fourth and sixth bundles
are pinned, then the lateral displacements at these locations
will be zero. Hence rows 4 and 6 of the displacement matrix
will contain zeros.
Similarly, for laterally rigid constraints, the response of
the string to forces p, and pfi will be zero. Therefore
columns 4 and 6 of Y(u)) will contain zeros. Because forces
p are non-zero, the fourth and sixth rows of H(w) must contain
zeros to balance the matrix equation (M153). The constraints
acting at the ends of bundles 4 and 6 cause forces p, and p,
to become redundant. Equation (M153) can therefore be reduced
to the form
'y
y
y
.y
21
31
51
y12y22
y32
y52
yl3y23
y33
y53
yis"y 2 5y35
y55-
«11
«21
«31
• « 5 1
«12
«22
«32
«52
«13
«23
«33
«5 3
«15
«25
«35
«55-
"Px0
0
.0
0
P20
0
0
0
P3
0
0
0
0
p5.(M154)
by eliminating the rows and columns corresponding to the constraint
locations.
In reduced form equation (M121) now becomes
. (p(u)))D (M155)
For K additional constraints along the fuel assembly, H'(to)
is of order (N-K, N-K).
Equations (M145) and (M150) in reduced form are
p'(ai) = (H'(w))"1 . Y'(u>) (M156)
and G' = (H')" 1 . G' . [(H'*)TJ~1 (M157)
<\, ^ <\, %
respectively.
A TIn general, matrices H1 (OJ) and (H1 (to)) will be non-singular
an 1 can b_e inverted. The resolved force functions p'(oj),
C'(w), f_' and ^ T^ n u i, are expanded in CALFOR to matricesF FM S r KMt>of order (Nxl) prior to output.
I »Cross-Dower matrices G-,(w) and G'(u) are printed with a message
to indicate which rows and columns have been eliminated.
10.3.2 Computation
Throughout subroutine CALFOR, the response matrices and force
equations are reduced if the ends of any bundles are constrained.
An example of the reduction of the receptance matrix is given
by statements CALFOR 109 and 110. Here, if the end of bundle p
is pinned, the p rows and columns of the matrix are erased.
In the special case of horizontally joined column vectors
(eg. DISMAT, of order (Nxm)),only row p is deleted. After
the calculation of the force vectors, FORVEC and AUTOPF
are expanded to order (Nxl) by inserting zeros in the place
of the redundant forces; refer to CALFOR 153 to 155 and 233 to
235.
-110-
11. LAMPS MATRIX PROCESSOR LANGUAGE
Because the computer modelling procedure involves numerous
matrix manipulations, a FORTRAN based matrix processor
language LAMPS was developed. LAMPS is a comprehensive
package of FORTRAN functions and subroutines designed speci-
fically for the execution of matrix operations. It facili-
tates orderly programming by allowing many matrix calculations
to be specified in a single program statement.
LAMPS automatically allocates storage space to matrices and
enables the user to clear the active portion of computer
core of unwanted information. This prevents core overflow
and avoids the need for time consuming overlay programs. The
matrix processor language runs under the CDC 6600 SCOPE
operating system at CRNL and has enhanced the rapid develop-
ment of the dynamic modelling work.
A typical example of the use of LAMPS is seen in the solution
of the matrix expression (M65), i.e.
BR z + ER . z = Ô (M65)
where BR =I ^ ^ \ and ER
The matrices BR and ER are constructed in statements
DYNMOD 392 and 393.
-111-
BRINV = ASSIGNC(INVERT(JOINV(J0INH(ZER0(N,N),A),JOINH(A,B))))
(DYNMOD392)
ER = ASSIGNC(JOINV(JOINH(NEG(A) , ZERO (N , N) JOINH (ZERO (N,N) ,C)))
(DYNMOD 393)
The square matrix U)= i*BR~ . ER is formed in DYNMOD 393.'K, O. <\j
W = ASSIGNC(MULT3(SCALAR(0.0,1.0),BRINV,ER))
(DYNMOD 394)
Equation (M65) is then solved to give the eigenvalues (MU)
and eigenvectors (MODES) of the fuel assembly.
MODES = ASSIGN(MODAL(W,MU)) (DYNMOD 395)
MU = ASSIGNC(MU) (DYNMOD 396)
In the above program statements, the following are LAMPS
functions: ASSIGN, ASSIGNC, INVERT, JOINH, JOINV, MODAL,
MULT3, SCALAR and ZERO. The matrix operations performed
by these functions are explained in APPENDIX V. The functions
can be embedded one within another, thus allowing a complete
matrix expression to be formulated in one program statement.
All matrices created by LAMPS are stored in a one-dimensional
array called SPACE. Complex number matrices are stored
column by column in SPACE and are each identified by a unique
integer name (a matrix pointer). In the example, the variables
A, B, BRINV, C, ER, MODES, MU and W are matrix pointers.
The pointers are not dimensioned and therefore LAMPS avoids
the use of subscripted variables.
-112-
Dynamic storage management in LAMPS allows the user to
continually optimize the available workspace. If matrices
are no longer required, they can be erased thus leaving room
for further matrix operations. This feature permits the
solution of large and complicated problems.
Sometimes, during execution of a program statement, a number
of unreferenced matrices are formed. Consider statement
DYNMOD 349, i.e.
QZO = ASSIGNC(MULT(D(L),ADD(D(SUB(MULT3(HALF,L,ADPDX),MULT
(ONËBOV(N),SMULT(L,SUB(FL,ADPDX))))),MULT(ONEBOV(N),D(SMULT
(L.FL)))))) .
(DYNMOD 349)
Here, QZO points to the matrix given by the operations specified
on the right-hand side of the program statement. However,
each embedded matrix operation will produce an unreferenced
matrix. Such matrices waste valuable core space and can be
deleted by using the CALL CLEANUP subroutine.
If a matrix is generated using an ASSIGN function, as in
statement DYNMOD 395, then it is placed in a 'protected'
state and is not erased by the CLEANUP subroutine. Therefore
by using the ASSIGN function and the CLEANUP subroutine the
workspace can be maximized and only important matrices stored.
The ASSIGNC function, seen in DYNMOD 392, 393, 394 and 396,
protects a matrix and deletes all unprotected matrices.
Matrices input through the free format LAMPS READn subroutine
are automatically stored in the protected state. A variety
of output functions is available; the majority print or plot
data in fixed format.
-113-
If errors occur during a run of the program (e.g. incompatible
matrix operations or SPACE overflow), these will be identified
by the comprehensive LAMPS debugging package.
The LAMPS matrix processor language is fully described in
Atomic Energy of Canada Limited report AECL-5977 "LAMPS: a
FORTRAN based matrix processor" by L. Kates and E. Reimer, (1976)
-114-
12. MEASUREMENT OF THE DYNAMIC MODELLING PARAMETERS
In order to construct an accurate dynamic model of a fuel
assembly, various structural and hydrodynamic modelling
parameters must be determined. Some of the constants used
in the model can be obtained from design specifications,
drawings or simple measurements, e.g. bundle masses, lengths,
element diameters, numbers of elements per bundle and hydraulic
diameters. Variables like fluid densities and velocities,
in single and two-phase flow, can be inferred from measure-
ments of the thermodynamic properties of the fluid.
However, certain modelling parameters must be evaluated by
experiment. These include structural and fluid damping,
friction and drag coefficients, dynamic stiffness and hydro-
dynamic mass. T' J methods used to determine these parameters
for CANDU-Li.iV reactor fuel are outlined below.
12.1 Structural Parameters
The bundle idealization in the computer model is arranged to
closely represent the fuel assembly of interest. For example,
the basic computer model of the vertical fuel string of the
CANDU-BLW reactor consists of 12 bundles standing above a
single end support. This idealizes the shield plug, spring
assembly and ten fuel bundles as shown in Figure 16.
Stiffnesses (K., , K and K ) and structural damping
factors (gCST» 8 E N D and gpAR) were obtained by examination
of the transfer function characteristics of the fuel assembly
in air. Figure 17 shows the experimental arrangement used to
measure the response spectra of an inverted CANDU-BLW reactor
fuel assembly (Ref. 5). Velocity transducers, located near
-115-
the bundle endplates, and an electrodynamic shaker were used
to measure the mobility and receptance between different
points on the fuel string over the frequency range of interest.
Using these -transfer function spectra, the natural frequencies
and mode shapes of the fuel assembly can be accurately
determined.
The modelling parameters K c g r K ^ , K p A R, g ^ , ^ and £ p A R
are then adjusted to give the best agreement between cal-
culated (using the computer model) and measured transfer
functions and modeshapes. The values that most accurately
represent the observed stiffness and structural damping forces
are then incorporated in the dynamic model of the fuel
assembly.
Examples of computer curves fitted to experimental data are
shown in Figures 18, 19, 20 and 21. The first shows the pre-
dicted and measured mobility curves of a full-length CANDU-
BLW reactor LS-3 fuel string. Figure 19 is a comparison of
experimentally measured and predicted modeshapes of a CANDU-
BLW reactor split-spacer fuel string. Examples of the
determination of structural stiffness and damping parameters
from tne vibration of component parts of a fuel assembly are
shown in Figures 20 and 21. Here, using equipment similar to
that shown in Figure 22, two-bundle and four-bundle structures
were driven at their mid-point to give the mobility curves of
Figures 20 and 21 respectively (Ref. 6).
End constraint stiffness and structural damping factors are
determined from measurements of the transfer functions of
the constrained or supported sections of the fuel assembly.
-116-
12.2 Fluid Parameters
Certain hydrodynamic parameters are best determined by examina-
tion of the motion of component parts of the fuel assembly
first in air and then in fluid.
12.2.1 Hydrodynamic Mass per Unit Length M and Viscous Drag Coefficient C.T
M and C of BLW reactor fuel bundles were found using the
experimental rig shown in Figure 22. This apparatus consists
of two empty fuel bundles clamped end to end at the mid -point
of a tensloned support rod. The rod ends are clamped on the
axis of a flow tube which replicates a BLW reactor pressure tubo .
The structure is harmonically excited at its mid-point by an
electrodynamic shaker. A piezo-electric force transducer
measures the applied force, and the response is monitored using
a velocity transducer also mounted at the mid-point of the
s tructure.
The two-bundle assembly behaves like a simple, damped mass-
spring system. It is forced to vibrate in air and in fluid
over a range of frequencies centred on the natural frequency.
Figure 23 shows typical driven-point receptance magnitude plots
for two Gentilly I fuel bundles (in a 105.9 mm ID test section)
in air and still water. The downward shift in the natural
frequency of the system caused by the hydrodynamic mass effect
is clearly seen. There is also a corresponding increase in
viscous damping when the system oscillates in fluid. Dis-
placement amplitudes were maintained at a constant RMS value
throughout the experiments.
a) The hydrodynamic mass per unit length M is determined using
the expression (Ref. 7)
mM = (m. + C 2 % +-*-) . \\^) - 1) (69)
-117-
where m, = bundle mass per unit length,
m = support rod mass per unit length,
m = mass of shaker spindle and transducers,S
L = l e n g t h of s u p p o r t r o d ,
CT and C, a r e c o n s t a n t s (C = 0 . 4 0 , C - 1 . 0 2 ) , and f and1 c. i- 2. na
f are the natural frequencies of the two-bundle assemblynw
in air and fluid (water or two phase) respectively.
The experimental results have shown that the hydrodynamic
mass per unit length of a fuel bundle in static single-phase
liquid can be predicted by the following formula.2
+ 1
M =1 — „ I - p D * (70)
where D is the pressure tube or Eest section internal dia-
meter, p is the fluid density and D is an effective bundlee
hydrodynamic diameter. D is defined by the empirical relation
De - (0.945) . (DB) . ~ (71)
where D is the outside diameter of the bundle, D is the
diameter of a single fuel element, and c is the average clearance
between elements.
kc is given by E n . C. (72)
1 = 1 x 1
-118-
Here, N is the total number of elements in the bundle, n.
is the number of elements in the i ring, C, is the clearance
between elements of the i ring and k is the number of rings
of elements. A plot of equation (70) is shown superimposed
on experimental results in Figure 24.
In air-water mixtures the fuel bundle hydrodynamlc mass
decreases linearly with increasing air volume fraction. The
rate of decrease of hydro-dynamic mass is found to be greater
than the rate of decrease of mixture density. Figure 25 shows
a plot of the experimental results for an 18-e.lement fuel bundle
over the air volume fraction range 0 to 40%.
b) The damping ratio ef of the assembly in fluid is calculated
from
(i) the 'sharpness' of the resonance peak in the
receptance magnitude plot, Af£f = - < 7 3 )
n
where Af is the resonance peak width at the half power
point and fn is the natural frequency, or
(ii) from the rate of change of phase angle between
force and displacement with frequency at resonance
e = — (74)
f (^nw ^df nw
-119-
If the fluid damping and structural damping forces are
additive, then the following relation is assumed to apply
ef = 2Ç + y (75)
Ç and Y are- the viscous and structural damping ratios of
the system; Y is assumed constant in air and fluid
(i.e., for two similar systems, one structurally damped
and the other fluid damped^oth with equal amplitudes at
resonance, the structural damping ratio of one is equal to
twice the fluid damping ratio of the other). Damping ratio
measurements in air give e = Y- Hence from measurements of
e and e_, Ç can be deduced,a f
The fluid damping ratio increases with increasing air volume
fraction. Experimentally determined damping ratios of three
different designs of CANDU fuel bundle are plotted against
air volume fraction in Figure 26. These results are averages
of the damping ratios obtained from experiments using a
variety of test section internal diameters.
Knowing the fluid damping ratio, the dynamic modelling parameter
C (metres/s) can be calculated from the work done by the
damping forces during one period of oscillation of the two
bundle system.
c) The viscous drag coefficient C is calculated from the fluid
damping ratio Ç as follows.
-120-
At resonance, the stiffness and inertia forces oppose one
another and the excitation force overcomes the damping
force. The excitation force and displacement are in quadrature;
therefore -the damping force will be in phase with the velocity.
With reference to Figure 27, the peak lateral velocity of
the slice of thickness fix, in the still fluid, is -r -(x) .3t
Hence, from equation 22, the peak viscous drag force on the
s l i c e i s given by
dF = - | pn D CD j £ (x) . <5x (76)
A plot of the viscous drag force and velocity for one oscil-
lation of the slice is shown in Figure 28. At the arbitrary
time t, the rate of dissipating energy by viscous damping is
given by
dF sinwt . -||(x) sinwt. (77)
Therefore, the energy expended per cycle is
T ^ ( x )
- d FdF . | T ( X ) . sin2wt . dt = — (78)2 f
o nw
The work done by the drag forces per cycle on all slices of
the two bundle assembly is, from (76) to (78),
nw
L
pn D CD f (|j(x))2 . dx (79)J
-121-
If the drag forces on the fuel bundles are much greater than
those on the support rod, then
WD " i f • Pn D CD I <^ ( x>>^ • dx <80>nw
Xl
The two-bundle assembly can then be considered to be an equi -
valent damped mass-spring system. The damping force per unit
ve loci ty of the simple system,C,is given by the expression
Ç . CCRIT (81),
where C_DT,_ is the critical damping value.
Using the relation
CCRIT » 4 * me fnw <82>
in equation (81), the peak viscous drag force becomes
DF = ** * me fnw < £
3yIn equation (83), "ât T /j ^s t ' l e Pea^ la teral velocity at the
mid-point of the two-bundle assembly (by definition, (•??•),/«ot L/2
is also the peak velocity of the equivalent mass-spring
system). m is the effective mass of the bundle system given
by
/m = / M(x) . U(x)) 2 . dx (84)6
-122-
where M(x) is the mass per unit length at x, and (j>(x) describes
the normalized mode shape of the bundle system at the natural
frequency f . If the mass per unit length of the support
rod is small' compared to that of the fuel bundles, then
rae - MF j (<|>(x))2 . dx (85)
where M^ is the mass per unit length of the fuel (including
hydrodynamic mass and rod mass). The work done by the drag
forces acting on the equivalent system is
(86)
nw
x2 ?
Hence wn = 2ir Ç ( f ^ )^ . Mw / (<Kx)) . dx (87)L/2
. M fF J
Equating equations (80) and (87) gives
P 2^ (d) (x) ) . dx
/ " 2 dï<-))2 •p n D
/ * 3W 9dx
However, at resonance, because (•gT)-»/? ^"s t * l e P e a^ velocity
along the two-bundle system,
-123-
*2 / a 2x . / ' *• J
1 . xi
Hence equation (88) reduces to
8TT f CM,,nw s F
C = (90)p n D
All the variables on the right-hand side of (90) can be
determined by measurement and experiment.
Hydrodynamic mass and fluid damping factors have been evaluated
using the above methods for a variety of CANDU-BLW reactor
fuel bundles in water and air-water mixtures in various sizes
of pressure tube. The r e su l t s are reported in de t a i l in Ref. 8.
12.2.2 Fr ic t ion Coefficient
Fluid shear forces at the surfaces of the flow channel and
fuel assembly cause a f r ic t ion pressure drop in the di rect ion
of flow. The f r ic t ion coeff icient can be calculated from
fr ic t ion pressure drop measurements.
In the dynamic model, the f r ic t ional pressure drop force per
unit length acting on a bundle is given by equation (26), i . e .
FF = 2 P D " 2 CF n D5
H
-124-
If A is the cross-sectional flow area between the bundleF
and pressure tube, and m, is the coolant mass flow rate,
then (91) becomes
In addition, the frictional pressure drop force
per unit length will be
( 9 3 )
where (-JT~)„ is the measured frictional pressure drop gradient
along the fuel-channel system, and A is the cross-sectional
area of the fuel bundle.
Equating equations (92) and (93) leaves the expression
c =F nD
The frictional pressure drop gradient is measured by experiment
for a variety of mass flow rates. The other variables in
equation (94) are calculated from design specifications. Hence,
the dimensionless factor C is determined. A more accurater
value for C will be obtained by direct measurement of the
longitudinal fluid friction forces acting on a fuel bundle.
12.2.3 Base Drag Coefficient, CgThe longitudinal base drag force acting at the end of the
fuel assembly is given by equation (30)
i-' FB " I Pend Deq Uend CB
-125-
An effective base drag coefficient can be obtained by
measuring the longitudinal pressure drop across the end of
the fuel assembly. If AP is the pressure drop (with s ta t ic
head subtracted), and A is the equivalent cross-sectional
area of the base of the assembly, then from equation (30)
A AP
CB = T 7 p 2 p22 end eq end
Here, the base drag coefficient takes into account the longi-
tudinal friction and velocity head pressure drop at the base
of the fuel string; i t therefore does not represent a genuine
form drag coefficient.
Because the computer model calculates the base drag force
from equation (30), the numerical value ascribed to Deq
can be arbitrary if it is first used in equation (31) for the
calculation of C .D
Form drag force measurements on fuel assembly end sections or
scale models will yield the most accurate value for the
base drag coefficient.
12.2.4 Free End Factor f
The lateral non-conservative force F acting at the free
end of the fuel assembly is given by equation (31), i . e .
FNC = " ( 1 " f ) Mend Uend « j f rend + "end V
where f is the end shape factor or free end factor.
Theoretical work has shown that the dynamic stabil i ty of
clamped free cylinders can depend strongly upon the parameter
-126-
f (Kef. 9). Similar analysis has shown that the magnitude
of f can dominate the stability of CANDU-BLW reactor fuel
strings (Ref. 1). Therefore, a computer model of a fuel
assembly having an unsupported end might only yield reliable
results if the factor f is known accurately.
If the end of a fuel assembly is supported, as in the clamped-
pinned configuration of the UKAEA-SGHWR design, the end force
becomes redundant and the stability of the fuel is independent
of f.
f can be determined by examining the stability characteristics
of a fuel assembly in parallel flow. After all other dynamic
modelling parameters have been determined, f can be adjusted
to give the optimum fit of computed results to expérimental data.
Alternatively similar techniques applied to the study of small
scale models or end sections of fuel assemblies will provide
a value for f. In the computer model of the CANDU-BLW reactor
fuel, the value f = 0.8 is often used.
-127-
13. DYNAMIC MODELLING EXAMPLES
This chapter describes an assortment of results obtained
using the dynamic modelling program DYNMOD. Detailed infor-
mation concerning the input and output format of the program
is presented in Atomic Energy of Canada Limited Report number
AECL-6068, 'DYNMOD: A Users' Manual'.
13.1 Natural Frequencies and Mode Shapes
All runs of DYNMOD produce a printout of the 2N natural
frequencies and 2N modeshapes of a N degree of freedom
structure. Figure 29 shows the natural frequencies (Hz),
grouped into modal pairs, and first 12 modeshapes (normalized
bundle end deflections) of a SGHWR fuel assembly in single
phase flow of 30 kg/s. The end-pinned fuel assembly was
idealized into 29 bundles; a table of the parameters used
in the dynamic model is shown in Figure 30.
At this mass flow rate, all the modes of vibration are stable
and damped. The negative imaginary parts of some of the
natural frequencies are a result of including structural
damping in the model. This was discussed in Section 3.3
and does not indicate the pteseace of instabilities.
The modeshapes describe wares travel} ing along the fuel
assembly. (The j eigenvector travelling wave of the
assembly in parallel flow sometimes bears little resemblance
to the j modeshape of the string in the absence of flow.)
Figure 31 shows the effect on the n;tural frequencies of
increasing the coolant mass flo'.' rate to 100 kg/s. It can
be seen that DYNMOD prints the eigenvalues in order of
increasing modulus, not in n u.ne rie .il order. In the first
and second pairs of frequencies, fhe magnitude of the
imaginary part is mucii greater than that of the real part.
Examination of the complete ^igenvector
-128-
(i.e.
V
shows that the velocities and displacements along the fuel
string for these modes are in phase. Therefore, at this flow
velocity, the fuel assembly exhibits buckling in the first
and second modes. The remaining natural modes of vibration
are stable and damped.
13.2 Travelling Waves
The motijn of a fuel assembly vibrating at a natural frequency
is described in generalized coordinates by equation (M66).
The natural frequencies and corresponding normalized deflec-
tions of an inverted BLW reactor fuel string are shown in
Figure 32; The structure was idealized into 12 bundles
and the parameters used in the dynamic model are shown in
Figure 33. Viscous drag forces and structural damping were
not included in the model. The damping seen in the natural
frequencies derives from the normal component of the fluid
friction forces acting on the string.
Figure 34 shows the first four modeshapes of the inverted
BLW reactor fuel assembly in parallel flow.
Using equation (M66) each modeshape was plotted after each
of ten equal increments of time covering one period of
oscillation of the string. The changing position of the
nodal points along the string indicates the presence of
travelling waves.
-129-
13.3 The Effect of Flow Rate on Fuel Assembly Stability
Usually, the natural frequencies of fuel assemblies in
parallel flow change when the fluid mass flow rate is altered.
An example of this effect is shown in Figures 35 and 36.
These plots show the frequency shift and stability limits
of the first three modes of vibration of an SGHWR fuel assembly
as the flow rate is increased. The modelling parameters used
to compute these results were the same as those tabulated in
Figure 30. However, in order to artificially reduce the
damping in the assembly, the structural damping factors have
been set to zero.
Each plot shows that the frequency of oscillation of the sys-
tem gradually decreases to zero with increasing mass flow
rate. The damping ratio, given by (y_/vi ), increases with
flow rate.
If the flow rate is increased beyond a certain value, the
fuel assembly becomes buckled in its first mode. The frequency
of oscillation drops to zero and the imaginary part of one of
the eigenvalues in the first modal pair assumes a large
negative value.
A further increase in flow rate causes buckling in the second
mode and eventually the third mode. Before buckling, however,
the third mode exhibits a fluid elastic instability. Here,
the imaginary parts of the pair of natural frequencies have
large negative values while the real parts indicate the
presence of low-frequency travelling waves. On reaching the
buckling flow rate of the third mode, the second mode will
have returned to a heavily damped state of vibration.
-130-
(As mentioned in Sections 3.3 and 13.1, the most reliable
method for differentiating between buckling and fluidelastic
instability is to examine the relative phase between velocity
and displacement at points on the fuel assembly for the mode
shape of interest.)
13.A Changes in the Hydrodynamic Parameters
Using the SGHWR fuel assembly first mode frequency against
flow rate plot of Section 13.3, the effects of large variations
in the fuel parameters C_, C_, C_, M and f were investigated.U D r
(The effects of the parameter changes on the higher modes is
similar but less dramatic.) Because the SGHWR fuel is
constrained by a piston seal at its lower end, the free end
factor f has no effect on stability.
The stability curve modifications corresponding to changes in
the remaining four parameters are shown in Figure 37. The
limits A and B in each curve describe the effects of arbitrarily
large variations in the modelling parameters.
It can be seen that the vibration frequency and critical mass
flow rate are most affected by changes in M and C . ChangesF
in CR have a large effect on the damping ratio of the system
below the c r i t ica l mass flow rate. Figure 37 illuminates the
relative insensit ivity of the SGHWR fuel stringer to large
changes in the base drag coefficient C_.
Computer experiments of this sort are used in the design of
dynamically stable fuel assemblies. For example, given a
particular coolant flow rate, i t may be desirable to induce
first-mode buckling of the SGHWR fuel assembly. Computations
suggest that the fuel string must have a low stiffness and
-131-
the fuel bundle should have a large hydrodynamic mass and/
or friction coefficient. In this simple case, such qualita-
tive suggestions will be obvious to a fuel designer. However,
the dynamic model predicts the optimum quantitative alterations
required in the fuel design; these changes are not obvious.
13.5 Natural Frequencies and Mode Shapes of Uniform Beams
The dynamic stability and vibration characteristics of integral
beams can be examined using the dynamic model. The beam is
idealized as an articulated string of equal length segments.
Beam stiffness is incorporated as
elK = ^ (Newton . metres)Co 1 Q
where el is the flexural rigidity of the beam and £„ is
the segment length, K is set to zero and K is set to
an arbitrarily high value (about 2 or 3 orders of magnitude
greater than K „,; if K p A R i-s s e t t o° large, scaling errors
will occur ) .
If the beam has a low shear stiffness it is best modelled
by settingci
with K C S T = 0. K p A R is set equal to K.A.G.g.«.fi (N.m)
where G is the modulus of rigidity, A is the sectional area
and K is a shape factor for the cross section.
Beam idealizations provide the most accurate results when the
length of the segment adjacent to an end constraint or a design
discontinuity is made equal to 1/2 £„. (The computer model
- 1 3 2 -
pred ic t s the first-mode frequency of a uniform can t i l ever beam
to within 0.01% accuracy using only a twelve segmentsidealiza-
t i on . This accuracy i s achieved because the segments masses
are d i s t r i b u t e d and not lumped.) With su i t ab le changes to
the vectors K g , ^"Nn a n d ^PAI i s i S P o s s i b l e t 0 model non-
uniform beams and components.
Figures 38 and 39 show the firs: : thr=-v; frequencies and mode-
shapes of, a uniform c a n t i l e v e r , a c i anpe. d-clamped beam, a
clamped-midpinned beam and a p j.nned - r inned-pinned beam. All
the ca lcu la t ions assume weight lass ind f lu id - f ree cond i t ions .
The f lexura l r i g i d i t y of each bea-n i s 5 x 10^ N.m , the beam
masses and lengths are 1100 kg and i l metres respec t ive ly
and twelve segments were used to idu.-^lize each case.
To demonstrate the basic accuracy of the model, the f i r s t
three na tu ra l frequencies computed for the can t i l eve r are
compared with t he i r analyt i^; i l values in the table below.
CANTILEVER
MODE
' MODE
t MODE1
1
2
3
ANALYTICALNATURAL FREQUENCY
0
0
1
1034 Hz
64 79
8147
COMPUTEDNATURAL FREQUENCY
Q
0
1
1034 Hz
6511
8307
% DISCREPANCY
0
0
0
.01
49
88
13.6 Transient Forced Response of a Uniform Cantilever
A single l a t e r a l harmonic force of amplitude 1 Newton and
frequency 0.1 Hz was applied to the t ip of the canti lever
shown in Figures 38 and 40.
The time domain response of the t ip of the canti lever is shown
in Case 1, Figure 40, where the time t = 0 corresponds to the
-133-
instant of application of the harmonic force. Two transient
response curves have been plotted. One is the response to a
sinusoidal force (i.e., first force peak of 1 N at 2.5 seconds),
the other is the response to a cosinusoidal force (i.e., a
force peak of 1 N at t = 0) . The amplitudes of the curves rise
with time to a peak steady state value. At steady state, the
response curves are TT/2 radians out of phase.
Case 2, Figure 40, shows a similar response curve but the-4forcing frequency of 10 Hz is well below the first natural
frequency. Here, the tip response is in phase with the
sinusoidal force. The computed amplitude of the response is
8.879 mm: this is in good agreement with the analytical
static deflection of 8.873 (0.034% discrepancy). The response
to the cosinusoidal force contains high frequency components.
This is because the cosine force peak is suddenly applied to
the cantilever tip at t = 0. Hence the response consists of
the forced low frequency part (10~* Hz) plus the high frequency
impulse components (natural frequencies).
Steady Harmonic Response
In the UKAEA - SGHWR, the fuel stringer is located in a vertical,
co-axial pressure tube/standpipe arrangement, see Figure 41.
Coolant is pumped upwards through the pressure tube and is
extracted from the standpipe via a riserpipe positioned above
the neutron shield plug. The riser axis is at 90° to the
standpipe axis. Therefore, there is a component of cross-flow
over the hanger bar near the riser pipe.
At typical single phase reactor flow rates, there will be a
steady fluid drag force on the hanger bar towards the riser
of about 10.6 Newtons. There could be a lateral harmonic force
-134-
on the hanger bar of about 3.5 Newtons peak amplitude at
21.7 Hz caused by vortex excitation.
The responses to the steady and harmonic forces are shown ir.
Figure 41. In each case the fluid forces have been resolve'.'
into two equal components acting above and below bundle 10.
A knowledge of the steady and harmonic response of the fuel
assembly to the cross-flow forces assists in the prediction
of pressure seal and fretting damage. The maximum steady
deflection towards the off-take pipe indicates the possibility
of contact between the fuel string and flow channel, and thf-
most probable contact point. For the fuel assembly described
in section 13.1, the most likely position of contact between
stringer and channel is at the fins of the neutron shield ping
(top curve, Figure 41). If the fuel stiffness were reduced
by a factor of 5, contact might occur at the second intermediate
grid - depending upon the design clearances along the string
(Figure 42).
Referring to Figure 41 for the stiffer fuel, it can be seen
that the motion of the hanger bar between the channel seal
and neutron shield plug is similar to the second mode stationary
oscillation of a clamped-pinned beam. Below the hanger bar
the motion is a lower amplitude travelling wave. Knowing the
contact pressure between the fuel string and the flow channel,
and the impulse and rubbing parameters, it will be possible
to assess the wear rate in the contact region.
The maximum bending moments and strains in the channel seal
plug can be calculated from the response characteristics of the
fuel assembly.
- 1 3 5 -
13.8 Transfer Functions
The transfer functions receptance and mobility are calculated
between points on the fuel assembly, with or without parallel
flow, by the subprogram RECMOB. Figure 43 shows the auto-
receptance at the mid-point of a simply supported beam. The
beam has the same dimensions and stiffness as those considered
in Section 13.5 and, in addition, a structural damping factor
of 0.1 has been introduced.
The second mode of vibration of the beam includes a node at the
mid-point of the beam. Therefore the second mode receptance
peak is absent in Figure 43. Damping has the effect of lowering
the receptance peak and smoothing the phase transition through
resonance.
Figure 18 shows the experimental application of the transfer
function subprogram.
13.9 Response to Random Forces
The spectral response of fuel assemblies to excitation of a
random or complicated nature can be calculated by subprogram
RANDOM. The response of the inverted BLW reactor fuel assembly,
described in section 13.2, to an array of fluid forces was
evaluated.
Each bundle end experiences a force. The time averaged power2
spectral densities (N /Hz) of the forces are described in
Figure 44 for the frequency range 1 to 10 Hz. Forces acting
at the ends of bundles 1 and 2 are correlated to the extent
shown by the cross-power spectral density curves, similarly
for the correlation between forces at bundles 11 and 12. The
forces acting at bundles 2 to 11 inclusive are uncorrelated
-136-
on average. The spectral response of the fuel assembly to
these forces is shown in Figure 45. Each curve shows the
auto-power spectral density of response at the ends of the
bundles for the range 1 to 10 Hz.
Curve R shows the root mean square displacement at the
bundle ends. Arrows mark the positions of the first six
natural frequencies of the fuel assembly in spectrum number
12. The third and fourth mode-shapes of the bundle string
are clearly depicted by the response spectra.
Resolved Force Calculations
If the response and transfer function characteristics of a
fuel assembly are known, then the nature of the forces,
resolved at the bundle ends, can be deduced using subprogram
CALFOR. In general, at a particular frequency, the power
spectral density matrix of response will identify a unique
p.s.d. matrix of force, via the transfer function matrix;
this was considered mathematically in chapter 10. A simple
example of the use of CALFOR is given below.
The vector of displacement amplitudes (metres) along the
simply supported beam of Section 13.8 is given by DISMAT in
Figure 46. It is assumed that the beam is vibrating at a
frequency of 1 Hz and the displacements are in phase.
2The intensity spectrum matrix of resolved forces (N ),
which caused the measured response is calculated using
CALFOR. The reduced power spectrum matrix is shown in
Figure 47. Because the response data took the form of a
discrete spectral line at each bundle end, a discrete force
spectrum can be calculated. These data are shown in Figure 48.
-137-
The results indicate that the applied harmonic load is a
uniform distribution of force represented by 1 Newton acting
at each unconstrained bundle end.
2Auto-power (intensity) spectrum of force (N ), mean square
and root mean square values of the forces at the bundle
ends are also supplied by CALFORJ see Figure 49. If the
measured displacement data had been in the form of power
spectral densities, then the forces at output would have
been power spectral densities (N /Hz).
-138-
14. CONCLUSION
Experiments have shown that the dynamic model will predict
the modeshapes and stability of continuously flexible
cantilevers in confined axial flow (ref. 10). Qualitative
results obtained from similar experiments on complete nuclear
fuel assemblies are in good agreement with theory (ref. 11).
This work has demonstrated the existence of waves travelling
along the fuel string as suggested by the model. Further
full-scale experiments, involving instrumented fuel assemblies
in single and two-phase flow, are being prepared. Data
from these experiments will also be used to check the validity
of the model.
A fuel designer can use the model to provide a qualitative
prediction of the dynamic characteristics of a proposed fuel
assembly. If the results are satisfactory, a prototype
assembly can be built and modelled accurately. It will then
be possible to investigate numerically the effects of design
changes, breakages.,, channel contacts, irradiation deforma-
tions, etc, on the vibration characteristics and stability of
the fuel assembly.
Future developments of the computer model will take into
account the dynamics of the flow channel. The basic theory
can then be extended to include the linear and non-linear
interactions between fuel and channel. Ultimately it should
be possible to predict the mechanical damage caused by excessivi
fuel assembly and channel vibrations.
-139-
15. REFERENCES
1. PAÏDOUSSIS, M.P. "Dynamics of Fuel Strings in Axial Flow".
Annals of Nuclear Energy, 1976, 3, 19-20.
2. TAYLOR, G.I. "Analysis of the Swimming of Long and Narrow
Animals". Proceedings of the Royal Society (London),
1952, 214(A), 158-183.
3. LIGHTHILL, M.J. "Note on the Swimming of Slender Fish".
Journal of Fluid Mechanics, 1960, 9, 305-317.
4. PAÏDOUSSIS, M.P. "Dynamics of Cylindrical Structures Sub-
jected to Axial Flow". Journal of Sound and Vibration,
1973, 29, 265-385.
5. CARLUCCI, L.N. and FORREST, C.F. "Experimental Determination
of Fuel String Dynamic Parameters". Private Communication.
6. CARLUCCI, L.N. Private communication, 1977.
7. CARLUCCI, L.N. "Hydrodynamic Mass and Fluid Damping of Rod
Bundles Vibrating in Confined Water and Air-water Mixtures".
Transactions of the 4th International Conference on Structural
Mechanics in Reactor Technology, 1977, D3/11.
8. CARLUCCI, L.N. "An Experimental Investigation of Bundle
Hydrodynamic Mass and Damping in Water and in Air-water Mixtures"
Private communication » 1976.
9. PAÏDOUSSIS, M.P. "Dynamics of Flexible Slender Cylinders in
Axial Flow. Part 1: Theory; Part 2: Experiments".
Journal of Fluid Mechanics, 1966, 26, 717-751.
-140-
10. PETTIGREW, M.J. and PAIDOUSSIS, M.P. "Dynamics and Stability
of Flexible Cylinders Subjected to Liquid and Two-phase
Axial Flow in Confined Annuli". Atomic Energy of Canada Limited
Report Number AECL-5502, 1976.
11. FORREST, C.F. and MONTI, P.J. "Vibration Measurements on a
String of CANDU Fuel Bundles in Adiabatic Steam Water Flow".
Transactions of the 4th International Conference on Structural
Mechanics in Reactor Technology. 1977, D3/12.
-141-
KENDTOP SUPPORT
GRAVITATIONALACCELERATION(FOR Y = + 1)
CHANNELWALL
CHANNELWALL
if).tit
.c!e , Fp(YP> AND
FLOWDIRECTION
FL(P).d£ ARE THE HYDRO-
DYNAMIC FORCES ACTING ONSLICE àÇ OF BUNDLE P.
A ( P ).de .AC) iS THE
PRESSURE DROP FORCE ACTINGON SLICE tlC .
FB AND FNC ARE THE HYDRO-DYNAMIC FORCES ACTING ATTHE END OF THE BUNDLESTRING.
FIG. 1: BUNDLE IDEALIZATION OF A NUCLEAR FUEL ASSEMBLY IN
AXIAL FLOW.
-142-
SUPPORT
FLOW DIRECTION
STRING AFTERDISPLACEMENT
THE HYDRODYNAMIC FORCESACTING ON THE FUEL STRINGARE SHOWN IN FIGURE 1
STRING BEFOREDISPLACEMENT
lN.i\ \ BUNDLE N-1
BUNDLE N
FIG 2 FUEL ASSEMBLY GEOMETRY BEFORE AND AFTER DISPLACINGBUNDLE 1 BY §0, RADIANS
r iBFofo
(-11.«!16<.(-20.41173( 20.41173
: ( 49.993R'
(-62.39244
( 1ft.117n.-l
: (-114.?76=( U4.27hS
(-134.9300
( 140.441?(-140.443-1
( ?19.aiS9(-2]9.i->ÇB( 327.4°?^
i til ,%HS*i
( -10» ' . '37
(-2S20.7J--
.".-!.'-. I-'in. «SSI*
.-.4I40?3?E-r ' l>
.-.4706714E-01)
. - .3»2R' 'n7f-n| )
, — B4«fi2ft^|p — ] )
• — »c ; ' t0Po?SE~Ol )
• — # P Q ? 3 7 1 ' J Ç — ft] )
; ; ; | | | |
• - .1»B"»1 l « . E - f ? )
1 - 4 . ((-h.l( h.l
i-ii.
—r^Trr
! 2?.
i 34.
( S?i
I -H ?
(-144
(-172( 172I |-'7
1 401(-401
II«-0773
4731»4/ II»1<=2?4
.1S«S
.1071
.307]
, - .73 f t6470E-02)
• - •^3 f t2413E-02). - .S312413E-0S),- .6SR93R4E-02), - .6S«93f l4E-0?). - •6506116E-02)
, - .6411141E-0?), - . ' . 4 P 1 1 4 i e - 0 2 ). - .7731620E-0?), - .7711A20E-02 l• - .7490961E-02). - . IftMU^hHt-Oa;), - .h0937P3E-02). - .60937H1F-0?), - .764362 l .F -02 ). - .764362SE-02).- .HS97S19E-02)
, - . « S 1 1 ) •' -0,-1,-.8SC.97 4 •.- -021.-.76S.5323Î -02), - . 7 6 « ' 5 3 ? 1 E : - O 2 ). - .371B74SE-02), - .371H74 [ î t -0?T-,--.33004e,F,T-02l.-.3300<.S«E-Ci?). - . 9 1 1 0 6 9 2 t - 0 3 ). - .9110692E-03)• - .1369271E-021
. - .1343H37E-02)
. - .1343K37E-02)
. - . a?0d5 i4E-03>
.- .n20H I i94E-03>
.- .73f . f tS51E-03)
. - . / . l h 6 S S I t - U J I
.-.sao?29^t-O3)
.-.S802295E-01)
.-.26B8311E-03),-.?6flH311F-01).-.6S739Mr-t'4)
.-.2422094S-0?)
.-.2422094F-02)
.-.1132069F-02)
( S60.2797
( -«3?.13°
FIGURE 3: Natural Frequencies and First 15Modeshanes of a SGHWR Fue] AssemblyIdealized into 29 Bundles. The FuelAssembly is Clamped at the Top andPinned at the Bottom.
DYNMOD. C . S . G . H . W . CREOL)nzrx:
I
-ror SoTTOM
-144-
NORHALIZFDFREQUENCYDIFFERENCE
DISPERSIONCAUSED BYSTRUCTURAL ,DAMPING(ANISOTROPYEFFECT)
TYPICAL LOWER ANDUPPER REACTOR
FLOW RATES
\
IV-
\\\\
INSTABILITY—m- ANDBUCKLING
10 20 30 50 60
FIG, 4
MASS FLOW RATE(kg/s)
FREQUENCY DISPERSION IN THE FIRST MODAL PAIR OF ASGHWR FUEL ASSEMBLY IN SINGLE PHASE FLOft
RECEPTANCE
8.0 — UNIFORM BEAMFORCE
AMPLITUDE(mm/N)
BEAM LENGTH = 11 m DISPLACEMENTMASS = 1100 kgElg = 5x104N-mz
L. 1<j0 RELATIVEPHASE (OARCI.
RELATIVEPHASECURVES
FREQUENCY SHIFTSDOWNUARDS ASVISCOUS DAMPINGINCREASES
1.0 —
RECEPTANCEAMPLITUDECURVES
0.20 0.22 0.24 0.2b 0.28 0.30 0.32 0.34 0.3b 0.38 0.40
FREQUENCY (Hz) *"
FIG, 5 THE EFFECT OF VISCOUS DAMPING ON THE AUTO-RECEPTANCE AT THE MID-POINTOF A SIMPLY SUPPORTED BEAM AS THE DRAG COEFFICIENT CD INCREASES, THERECEPTANCE PEAK SHIFTS DOWNWARDS IN FREQUENCY
I
FIG. 6RECEPTANCE
8.0 —
7.0 —
AMPLITUDE(mm/N)
b.O
5.0
4.0
3.0
2.0
1.0
— 180
— 160
— 140
- 120
- 100
- 80
- bO
- 40
- 20
- 0
THE AUTO RECEPTANCE AT THE MID-POINT OF ASTRUCTURALLY DAMPED, SIMPLY SUPPORTED BEAMAS THE STRUCTURAL DAMPING COEFFICIENT gINCREASES, THE RECEPTANCE PEAK SHIFTSUPWARDS IN FREQUENCY
RELATIVEPHASE (°ARC)
BEAM LENGTH - 11m ,, DISPLACEMENTMASS = 1100 k g T
Elg = 5x1O4N-m2-
UNIFORM BEAM
FORCE
RELATIVEPHASECURVES
FREQUENCY SHIFTS UPWARDS SLIGHTLYAS STRUCTURAL DAMPING INCREASES
F I R S TN A T U R A L
FREQUENCY
RECEPTANCEAMPLITUDECURVES
i I i0.20 0.22 0.24 0.26 0.28 0.30 0.32
FREQUENCY (Hz)
0.34 0.36 0.38 0.40
SUPPORT
"•* T A H -2
END-PINNED.FREE TO MOVEVERTICALLY
FIG. 7 END-PINNED FUEL ASSEMBLY
SUPPORT
•*-« MID-PINNED.FREE TO MOVE
^+" VERTICALLY
FIG, 8 MID-PINNED FUEL ASSEMBLY
SUPPORT SUPPORT
MID-PINNED.FREE TO MOVE
'*•" VERTICALLY
u.END-PINNED.FREE TO MOVEVERTICALLY
REST POSITION OFFUEL ASSEMBLY
N-3
00I
FIG. 9 MID-AND END-PINNED FUELASSEMBLY
FIG. 10 THE FORCE VECTOR f ACTING ONTHE FUEL ASSEMBLY
-149-
SUPPORT
FUEL ASSEMBLY
f = p * exp(i w t )
EACH SPECTRAL LINE CONTAINSAMPLITUDE AND RELATIVEPHASE INFORMATION
AMPLITUDE
i l 1 I• 1
1
' I1 1111
1111
i
i l
, ^RELATIVE
. y PHASE
i l il ,1FREQUENCY ( u) ' '27
ili
i
i
it
ii il
•2 77-
N-2
Hi\ I I1 ' i!! Il \
I i
t li \\ ft 1127T
BANWID w 2tr
FMIN
FIG 1! DISCRETE SPECTRA FORM OF THE LATERAL FORCES ACTINGON THE FUEL ASSEMBLY
SUPPORT
BUNDLE 3
FIG,
(AUTO P.S D.RESPONSE)
(AUTO P.S.O.RESPONSE)
12 EQUIVALENT FORCES P(cu)ACTING ON THE FUEL ASSEMBLYPRODUCE THE COMPLEX DISPLACEMENTS
AUTO P.S.D OF FORCEACTING AT ENDOF BUNDLE I
Fil
GF,,2(REAL)
BUNDLE 1
ai
U12
(IMAGINARY)
CROSS P.S.D. OF FORCESACTING AT ENDS OFBUNDLES I AND 2
2-rr
F22
(NOTE: G Frs
AUTO P.S.D. OF FORCE ACTINGAT END OF BUNDLE 2
CROSS P.S.D. OF FORCESACTING AT ENDS OFBUNDLES 2 AND 1
FMIN
. FIG, 13 NATURE OF THE POWER SPECTRAL DENSITY OFFORCE DATA USED IN DYNMOD
oI
G F U 1 (w)
FU3
FM INBANWIC
J FU1
20
10
AUTO P . S . D . FORCE SPECTRA
(ARBITRARY UNITS)
J FU2
20
10
1 2 3 4 5 6 7 8 9 10 11 12FREQUENCY
krrfTîlTrm1 2 3 4 5 6 7 8 9 10 11 12
JFU3
20
10
0
FREQUENCY ( o> /2w)
-nrrîiiTFREQUENCYP O I N T , m
UNCORF
HORiZONTHLLYIOIHED VECTORS
OF UNCORREUTEDP.S.D. FORCE
Tîpîl ( w )
1
21
6
5
2
16
b
5
3
13
7
b
4
15
10
7
5
19
14
8
6
20
17
10
7
13
14
12
8
10
11
15
9
10
8
18
10
11
7
16
11
7
6
11
12
3
5
7
1 2 3 4 5 6 7 8 9 10 1 1 1 2
FREOUENCY ( < u , 2 7 r )
i
I
FIG. 14 SIMPLE EXAMPLE OF THE CONSTRUCTION OF THE (Nxm) UNCORRELATED FORCE MATRIXUNCORF FOR A THREE BUNDLE STRING AND TWELVE FREQUENCY POINTS ( L e , N = 3 , m = 12)
-152-
FUELASSEMBLY
MID-PINNED.FREE TO MOVEVERTICALLY
END-PINNED.-FREE TO MOVEVERTICALLY
FIG 15 FORCES ACTING ON A CONSTRAINED STRING OF SIXFUEL BUNDLES
-15 3-
OUTLET
GRAVITATIONALACCELERATION
FLOUCHANNEL
12• •
11=13=
10• •
9zrxz
8-"—
7=o=
633
5
4=cr
3
FLOU
DIRECTION
FUEL
BUNDLE
- SPRINGASSEMBLY
NEUTRON
SHIELD
PLUG
UATER INLET
FIG 16 A TWELVE BUNDLE IDEALIZATION OF A CANDU-BL*REACTOR FUEL ASSEMBLY
-15 4-
STRING SUPPORT
SHIELD PLUG
STRONG BACK
FUEL STRING
FUEL eUNOLE
VELOCITY TRANSDUCERS
LOAD CELL
F I G . 17 EXPERIMENTAL ARRANGEMENT USED TO MEASURE THE RESPONSE
SPECTRA OF At'. INVERTED CANDU-BLW REACTOR FUEL ASSEMBLY.
-155-
10"
10"
MEASURED
DYNMOD PREDICTION
10-5
FREQUENCY ( H z )
10 20 35
FIG. 1 8 : COMPARISON OF THE MEASURED AND PREDICTED MOBILITIES
OF AN INVERTED CANDU-BLW REACTOR FUEL ASSEMBLY.
( C a r l u c c i L . N . , P e r s o n a l C o m m u n i c a t i o n ) .
- 1 5 6 -
NORMALIZED DISPLACEMENT
MODE 2 MODE 3 MODE 4
MODE 6 MODE 8 MODE 9
F I G . 1 9 : COMPARISON OF SOME MEASURED AND PREDICTED MODESHAPES
OF AN INVERTED CANDU-BLW REACTOR FUEL ASSEMBLY.
- 1 5 7 -
10-2
TO"3
10-4
-
KCST
KPflR
KENO
GCST
GPflR
GEND
1 1
1 1
1
MEASURED
DYNMOD
= 2600 N-m/ rad
= 8700 "
= 1800
= 0.00= 0.09= 1.38
/
,y/i i i i
/
i i i 11
V1 1 1 1 1 1
10FREQUENCY (Hz)
FIG. 20 DRIVEN POINT MOBILITY OF A TWO BUNOLE ASSEMBLY
-15 8-
10-2
DYNMOD
KCST = 1OOO N-m/radKPftR = 7000KEND = 105 "GCST = 0.0GPAR = 0.3GEND =0.5
FREQUENCY (Hz)
FIG. 21 DRIVEN POINT MOBILITY OF A FOUR BUNDLE ASSEMBLY
- 1.5 9 -
NATURAL..C I R C U L A T I O N ^
RETURN
COMPLIANCEMAGNITUDEPHASEANGLE' FREQUENCY
TWO BUNDLE ASSEMBLY
SHOP AIR
FIG. 22: EXPERIMENTAL RIG USED FOR THE MEASUREMENT OF
THE HYDRODYNAM1C MASS AND DRAG COEFFICIENT OF
CANDU-BLW REACTOR FUEL BUNDLES.
- 1 6 0 -
100
AIR
STILL
««TER
•
*
o
FORCE(H)
3.1
1.6
3.1
S.I
1 «SSEUBLV IN AIRI1
FIG. 2 3: DRTVRN POINT RECF.PTANCF. MAGNITUDE PLOTS OBTAINED
FROM THE APPARATUS SHOWN IN FIG. ? 2 .
10
-
_
-
—4/w
MagnitudeData
O
oA
1 1
KPhaseData
©
•
i i
r
m
"" TTOD4
1 '
h2
e
Bundle Type
1828
37
Element
Element
Element
1
l
e
* ' •
- \
i i
1
1
19
De =
i
i
0.945
i
1
DBD
1
A • — — .
i
i i
-
-
- * — -
-
i i
5 _
1.2 1.3 1.4
FIG. 24: HYDRODYNAMIC MASS VERSUS BUNDLE AND TEST SECTION GEOMITRY.
-162-
t 0.6
0.4
0.2OOA
Proportional ToMixture Density
Test ChannelI.D. (mm)
103.4105.9108.0
, I10 20
Air Volume Fraction,
30 40
. 25: HYDRODYNAMIC MASS VERSUS AIR VOLUME FRACTION FORAN 18 ELEMENT BUNDLE.
-163-
0.2
O•r—
Rat
c
Q.
10O
'r— 0 1
a-
(
L
0
i
-
;—
-
— / /
/ / ^
1
1
l
^ —
O
OA
1
i
_s
—A
.——
O
Bundle Type
18 Element
28 Element
37 Element
i
1
"o—
A
-
110 20
Air Volume Fraction (%)
30
FIG. 26: COMPARISON OF TWO-PHASE DAMPING FOR THREE BUNDLE TYPES.
.. x
L ZÂ
1 —
DRAG FORCEAND VELOCITY
TAPPLIED FORCE
FUELBUNDLES
SUPPORTROD
TIME
FIG. 27 FIG, 28
-165-
FIG. 29: NATURAL FREQUEN
CIES AND NORMALIZED
BUNDLE END DEFLECTIONS
OF A SGHWR FUEL ASSEMBLY
IN SINGLE PHASE FLOW OF
30 kg/s.
2Jât>*7jL-ai.'.<::iI*3-<QZJ W u 9 m - J l . - . JMJ5I.U
ia7oa^ « «ÏDVIIOOoj*o-«7fc-#t« .21£?7frt>
lolLL-ÛI» (-.lJïa^i<.E-ûl.-.'^&^*•^lO7e74t>i I "3(-.bi .ci ïCî '*E-i l , - ,?*o6i 'Jl6<ftï > uit-.±rWie*>7E-tli-*2?«l*«V
J7E-II1I l-.^t.t.<.«3e-01.-.e7Jt^t2t-01dot-0 11 l - . l o i ûfc*i I t - 0 l , - .2o JdD»*E-Ûl
iJs«tillU-01.-.^«2tr/70L-Ull «-. J10ÏU/at- t t , - .^2H67^i t -»HS12b«bJL-ul.-.Ja>jrj«St-ail (- .y j(>i .«72t-Bl,- . i lJ lJ19t-BIJ
be74l7ilL-lJl. .ittalùHit-aii t .fJ^fc&tbE-dl* .£l*hlStL-ait
BUNDL1NUMBEI
1
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
; LENGTfI (m)
0.311
0.622
0.622
0.622
0.622
0.622
0.622
0.622
0.622
0.622
0-32!»0.U26
0.3ÏS
0.631
0.Ï2 8
0.025
0.176
0.J5-/
0.353
0.378
0.378
0.378
0.378
0.378
0.378
0.378
0.378
0.378
0.118
STRUCTURAL PARAMETERS
«ASS(kg)
3.809
7.620
7.620
7.620
7.620
7.620
7.620
7.620
7,620
7.620
3.970
15.764
4.082
7.830
4.070
13.326
2.310
4.575
13.431
24.964
24.964
24.964
24.064
24.964
24.964
24.964
24.964
24.734
8.628
NEL
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
64*
64*
64*
64*it
6464*
64
64*
64*
64*
1
DIAM(m)
0.0603
0.0603
0.0603
0.0603
0.0603
0.0603
0.0603
0.0603
0.0603
0.0603
0.0603
0.0802
0.0953
0.0900
0.0900
0.0900
0.0752
0.0691
0.0122
0.0122
0.0122
0.0122
0.0122
0.0122
0.0122
0.0122
0.0122
0.0122
0.1122
^CST ITMT»
(N.m) (N.m)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
10 7
1.389x105
1.389x105
1.389xlO5
1.389x105
1.389x105
1.389x105
1.389x10s
1.389x105
1.389x10s
1.389x10s
3. 323x106
3.323xlO6
1.369x105
1.369x105
3.456xlO6
3.456xlO6
10 7
1 0 '
7.978x10"
7.978x10*
7.978x10"
7.978x10"
7.978x10"
7.978x10"
7.978x10"
7.978x10"
7.978x10"
1.264x10"
KPAR(N.m)
10 7
1 0 '
10 7
1 0 7
10 7
1 0 '
10 7
10 7
10 7
10 7
1 0 '
10 7
10 7
10 7
10 7
1 0 '
10 7
1 0 '
10 7
1 0 '
10 7
10 7
1 0 '
10 7
10 7
1 0 '
10 7
10 7
1 0 '
GCST
o •
0
o0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
G
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.005
GPAR
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
OTHERPARAMETERS
Y=l
C = 9 . 8 0 6
(m/s2)
DCTOPX=0.0
DETOPX=0.0
RHO(kg/m3)
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
748.5
U(m/s)
0 . 0
0 . 0
0 . 0
0 . 0
0 . 0
0 . 0
0 . 0
0 . 0
0 . 0
-1.887
-3.775
-4.756
-6.313
-5.632
-5.632
-5.632
-4.519
-4.124
-6.469
-6.469
-6.469
-6.469
-6.469
-6.469
-6.469
-6.469
-6.469
-6.469
-4.298
CD(tn/s)
0.053
0.053
0.053
0.053
0.053
0.053
0.053
0.053
0.053
0.053
0.053
0.040
0.035
0.036
0.036
0.036
0.043
0.048
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.029
FLUID
CF
-0.035
-0.035
-0.035
-0.035
-0.035
-0.035
-0.035
-O..O35
-0.035
-0.035
-0.035
-0.028
-0 .033
-0.025
-0.025
-0.025
-0.029
-0.032
-0.022
-0.022
-0.022
-0.022
-0.022
-0.022
-0.022
-0.022
-0.022
-0.022
-0.015
PARAMETERS
CB
-1 .13
-1.1.3
-1.13
-1.13
-1 .13
-1 .13
-1 .13
-1.13
-1 .13
-1.13
-1 .13
-1 .13
-1 .13
-1 .13
-1 .13
-1 .13
-1 .13
-1 .13
-1.13
-1.13
-1 .13
-1 .13
-1 .13
-1.13
-1 .13
- 1 . 1 3
-1 .13
-1.13
-1 .13
DFQ
(m)
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
0.1033
FEF
0 . 8
n.s0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
0 . 8
DIAMH
(m)
0.07066
0.07066
0.07066
0.07066
0.07066
0.07066
0.07066
0.07066
0.07066
0.07066
0.07066
0.0508
0.02841
0.0410
0.0410
0.0410
0.05583
0.05803
0.00867
0.00867
0.00867
0.00867
0.00867
0.00867
0.00867
0.00867
0.00867
0.00867
0.07305
VM
(kg)
1.024
2.047
2.047
2.047
2.047
2.047
2.047
2.047
2.047
2.047
1.066
0.215
4.367
8.377
4.354
0.332
1.458
1.272
9.495
10.167
10.167
10.167
10.167
10.167
10.167
10.167
10.167
10.167
75.638
* ( 6 S n a r n e t u b e s * 3 f u e l p i n s )
FIG. 30: STRUCTURAL AND FLUID PARAMETERS USED IN A SGHWR FUEL ASSEMBLY PROTOTYPE
DYNAMIC MODEL.
W Pu
(** w
Pi <Cg 33
W tu
< o
en ZM MM
W ça
en
DO
Ë
-167-
I
r - i
FIG
.
S: ^
îfo 2£
i•
Î:• T 1 ' '
i' :
•; i
I
; [
iï
i
t r -
j
ok
ii ! •; ; 1
• i ;
1
p-
|
!
lo oi l l 'Ul UJ I
i t i i i i i l i; t» f » » • • • • • • • • n » • » » , . + ..
CO rf « . • i H H *
J..
iv ne -v
^ . . . . ._> L...
I II l]l II i i i i
u
-168-
(-1 .10)1)73 . .'*?S1T?
( •U.??*4<1 . ,]?S7<;4|
1-113.06?? . .1915J9Q( 11.1. MIR . .l«Jmi9«
-L. 1AUA*z$_ t «196BAAA .
3.1&BT13
.2?*?Wl'i
-20QÈÉLÛ7E-0J)
*3132EUE-nL]
FIG. 32: NATURAL FREQUENCIES
AND NORMALIZED BUNDLE END
DEFLECTIONS OF AN INVERTED
CANDU-BLW REACTOR FUEL
ASSEMBLY.
( .777938ÎE-0J. 1.45?1?«
I - . ? 1 9 * T ; S , . 1^3371
.1163TBB
•,1?1?613
t »il137^4f*01t (^30^9ft?€'—01) C*>»llfl)ftl^F.*nit a37P?3?SF.~01) I*»3R1^AS1E**D1* »1 O ft%D Xf **011
1 !j59.64<>7£-t}l. lfc9t630*t-QU 1 .1935Û72E-H1.-1(I321Q46E-Û1 ) 1 Iâ375DaSE-Ûl*»Il6B6Z9«E-ûi i.33b3D36£.-01
..i.-.flSafl9**E-01«i« . |
.(-.TtS09_4.4E-Hl..-,59?llSlE-fl21A3' 1 1 0 9 9 E 0 1
BUNDLENUMBER
1
2
3
u
5
6
7
8
9
10
11
12
LENGTH(m)
1.700
0.427
0.500
0.500
0.500
0.500
0.5.00
0.500
0.500
0.500
0.500
0.500
STRUCTURAL PARAMETERS
MASS
(ks)
42.0
8 . 6
26.7
26.7
25.7
26.7
16.7
26.7
26.7
26.7
26.7
26.7
NEL
1
19
19
19
19
19
19
19
19
19
19
19
DIAM(m)
0.08
0.0197
0.0197
0.0197
0.0197
0.0197
0.0197
0.0197
'").0197
0.0197
0.0197
0.0197
KCST(N.m)
0 . 0
0 . 0
5000.0
5000.0
5000.0
5000.0
5000.0
5000.0
5000.0
5000.0
5000.0
5000.0
KEND(N.m)
0.0
0.0
io'
106
1 0 '
io6
105
10 '
1 0 '
W
io'-
KPAR(N.ra)
1 0 '
10"
10*
10«
10*
10"
1 0 '
10*
10"
1 0 "
io-
1 0 *
GCST
Q
0
0
0
0
0
0
0
0
u
0
0
"EHD
0
0
0
0
0
0
0
0
0
0
0
0
GPAR
0
0
0
0
0
0
0
0
0
0
0
0
OTHERPARAMETERS
G=9.80(m/ s2)
DCT0PX=0.0
DETOPX'0.0
RHO(kg/m3)
770.0
770.0
770.0
760.0
740.0
550.0
420.0
350.0
300.0
275.0
255.0
240.0
U(m/s)
4.99
5.88
5.88
5.96
6.13
8.24
11.65
12.95
15.11
16.48
17.78
18.89
CD(m/s)
0 . 0
0 .0
0.0
0.0
0.0
0 .0
0.0
0 .0
0.0
0 .0
0 .0
0.0
FLUID
CF
0.003
0.003
0.003
0.00 3
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
PARAMETERS
CB
0.1.0
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
DEq
(m)
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
F E F
0 . 8
0 . 8
0.8
0.6
0 .8
0 . 8
0 . 8
0 .8
0 . 8
0 . 8
0 .8
0 . 8
DIAMH
(«0
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0,006
0.006
0.006
0.006
VM
(kg)
38.00
9.50
11.15
11.05
10.75
8.00
6.10
5.05
4.35
4.00
3.70
3.50
FIG. 3 3 : STRUCTURAL AND FLUID PARAMETERS USED IN A DYNAMIC MODEL OF AN INVERTED
CANDU-BLW REACTOR FUEL ASSEMBLY.
- 1 7 0 -
MODE
NORMALIZEDDISPLACEMENT
I I
:6•'• 5
'7. 4a
3• • « »
MODE 2
J 2 3DISTANCE ALONGBUNDLE STRINGFROM TOP SUPPORT (m)
MODE 3 MODE 4
FIG. 34: FIRST FOUR MODESHAPES OF AN INVERTED CANDU-BLW REACTOR FUEL
ASSEMBLY IN AXIAL FLOW. THE STRING DEFLECTIONS ARE PLOTTED
AFTER EACH OF TEN EQUAL INTERVALS OF TIME COVERING ONE PERIOD
OF OSCILLATION.
-171-
SGHHR FUEL STRINGER:IDEALIZED FOR USE IN THE DYNAMIC MODELLING
PROGRAM DYNMOD
FLOW CHANNEL
] COUPLING
COUPLING
G G (
Cj/1,9'G \J
20 21
g GRAVITATIONAL
; G (
22 23
ACCELERATION
! G G G
24
FUEL ASSEMBLY
25 26 27
FLOW DIRECTION
BOTTOM
3 G >
28
u
29
J
PISTON SEAL
G INTERMEDIATE GRID POSITIONS
z
SGHWR COMPLEX FREQUENCY AGAINST
FLO* RATE
FIRST PAIR OF NATURAL FREQUENCIES
(FIRST MODAL PAIR)
BUCKLfD
t * — y y y70 80 90 100 110 120 130 140 130
MASS FLOW RATE
(kg/s)
I MAG,
x REAL PARTS
• IMAG. PARTS
FIG. 35 : IDEALIZED SGHWR FUEL ASSEMBLY AND FIRST MODE STABILITY CURVE.
- 1 7 2 -
ELUOC
SGHHR
COMPLEX F R E Q U E N C Y A G A I N S T FLOW RATE
SECOND PAIR OF NATURAL FREQUENCIES
(SECOND MODAL PAIR)
10 20 30 40 50 60 70
150 MASS FLOW RATE
(kg/s)
* REAL PARTS
• IMAG. PARTS
SGHWR
COMPLEX FREQUENCY A G A I N S T FLOW R A T E
THIRD PAIR OF NATURAL FREQUENCIES
(THIRD MODAL PAIR)
10 20 30 40 50 60 70 80 90
1 MASS FLOW RATE(kg/s)
• 6 1 -
x REAL PARTS
• IMAG. PARTS
F I G . 36 : SECOND AND THIRD MODE STABILITY CURVES OF A SGHWR FUEL ASSEMBLY.
-17.3-
VARY CB
NOMINAL VALUE - 1 . 1 3
VARY CD
NOMINAL VALUE 0 .22 m/s
0 70MÂITFLÔW RATE
(kg/s)IMA6,
LIMIT A, CB = + 1.0LIMIT B, CB = -3.0
NEGLIGIBLE EFFECT ON DAMPING
MASS FLOW RATE(kg/s)
IMAG,
LIMIT A,CD = 0.0 m/sLIMIT B.CD = 0.3 m/s
VARY CF
NOMINAL VALUE -0.022
MASS FLOW RATE(kg/s)
IMAG,
LIMIT A,CF = -0.01LIMIT B,CF = -0.03
VARY VH
NOMINAL VALUE 10.17 kg(VM = HYDRODYNAMIC MASS PER
BUNDLE)
MASS FLOW RATE(kg/s)
IMAG,
LIMIT A. VM = 5 .0 kgLIMIT B, VM -- 15.0 kg
F I G . 3 7 : THE EFFECT OF LARGE VARIATIONS IN THE PARAMETERS C_, C_ , CD r D
AND M, OF THE FUEL, ON THE FIRST MODE STABILITY OF A SGHWR
FUEL STRINGER.
- 1 7 4 -
UNiFORM CANTILEVER
0.1034 Hz
1 2 3 4 5 6 7 8 9 10 11 12 BUNDLE
0.6511 Hz
i 1 1 1 1 11 2 3 4 5 6 7 8 9
1 .8307 Hz
1 2 3 4 5 64
UNIFORM CLAMPED-CLAMPED BEAM0.6641 Hz
1 2 3 4 5 6 7 8
NUMBER
1.8468 Hz
3.6500 Hz
FIG 38 FIRST THREE MODESHAPES OF A UNIFORM CANTILEVER ANDCLAMPED-CLAMPED BEAM
- 1 7 5 -
UNIFORM CLAMPED-PINNED-FREE BEAM
0.1805 Hz
I I I i —1 I 1 1 1 1 H1 2 3 4 i 5 6 7 8 9 10 11 12 BUNDLE
' NUMBER
1.1826 Hz
^
11 12
11 1.2
3.3613 Hz
1 2 3
UNIFORM PINNED-PINNED-PINNED BEAM
1.1610 Hz
| BUNDLE' NUMBER
1.7393 Hz
4.6371 Hz
FIG, 39 FIRST THREE MODESHAPES OF A UNIFORM CLAMPED-PINNED-FREEBEAM AMD A PINNED-PINNED- PINNED BEAM
- 1 7 6 -
1DEALIZED UNIFORM CANTILEVER
' i 2 1 3 1 4 1 5 1 s 17 i a r r
50
40
30
20
to
0
-10
-20
-30
-40
-50
<j LENGTH = 11 mMASS = 1100 kg
C A S E I EI,.5«IO«N.>»
FORCING FREQUENCY NEAR FIRST RESONANCE
F = IN, f = O.I Hz
* (FIRST NATURAL FREQUENCY = 0.1034 Hz)
FORCE
DISPLACEMENT
Fcos2 i r f t
D 1 TIME(SECONDS)
20
16
12
C A S E 2FORCING FREQUENCY WELL BELOW FIRST RESONANCE
" F = I N . f = IO"«Hz
TIME(> 103SECONDS)
CURVES S AND C ARE THE TIME DOMAIN RESPONSES TO SINUSOIDAL AND COSINUSOIOALFORCES Fs AND Fc RESPECTIVELY.
IN CASE 2 , THE HIGH FREQUENCY COMPONENT IN CURVE C IS CAUSED BY THE COSINEIMPULSE AT t = 0.FOR A TWELVE BUNDLE IDEALIZATION, THE DISCREPANCY BETWEEN THE ANALYTICAL ANDCOMPUTED STEADY AMPLITUDES IS LESS THAN 0.02%
FIG. 4 0 : TRANSIENT RESPONSE OF A UNIFORM CANTILEVER TO HARMONIC FORCES
APPLIED AT THE TIP.
- 1 7 7 -
IDEALIZED SGHWR FUEL STRING
FLOW CHANNEL
FLOW
S3
1
POINTS OF APPLICATIONOF CROSS FLOW FORCES
GRAVITATIONAL ACCELERATION
RESPONSE TO STEADY ORAG FORCES OF10.6 N. AT FLOW RATE OF 27.7 k g / s
BUNDLE NUMBER
•Or
-360
•320
-300
.9Qfl«HI
-260
-240 ,
•22o/
• /
1•160
•140
-120
• 100
> anou
•60
-40
-20
• 0
/ ,- / - • • •
/
/
/
I///
2 3
K
e
UJ_ i
ANG
IH
ASE
Q_
UJ
LA
TI
\ Ai 1 / 1\ » / \
\ » / \
\ \ / \\ \ / \\ w \
\ / \ /
\ 11 T'
4 5 4 7 _ ^ '
STEADY HARMONIC RESPONSE TO
EXCITATION OF 3 . 5 N AT 21.7
FLOW RATE = 27 .7 k g / s
x AMPLITUDE
• RELATIVE PHASE
r \\
9 13
VORTEX
Hz
\
\\
\i5 '
\>
\\
\\\
\
A n\ /V
19 20 22
V
\
\1
»1I
\\11
1II1
\' ,
V/!\211\1
"•I
Vt 26
BUNDLE
V.
\
111
I11
111
1
1
AfI28
1
NUMBER
FIG. 41: STEADY RESPONSE OF A SGHWR FUEL STRINGER TO CROSS-FLOW FORCES.
HANGER BAR
NEUTRONSHIELDPLUG FUEL BUNDLE
10
9
8
LATERALDEFLECTION
_ OF FUELSTRINGER
_ (mm)
DEFLECTION CURVEOBTAINED BY REDUCINGFUEL STIFFNESS BY A
FACTOR OF 5
EQUIVALENT DRAG FORCESAPPLIED AT TOP AND BOTTOMOF BUNDLE 10
DISTANCE ALONGFUEL STRINGER(METRES)
FIG, 42 STATIC DEFLECTION OF A SGHWR FUEL ASSEMBLY TOWARDS RISER PIPECAUSED BY A CROSS FLOW OF 27.7 kg/s
- 1 7 9 -
FORCE
RECEPTANCE(mm/N)
UN I FORMBEAM
DISPLACEMENT
PHASEDYNMOD PREDICTS THE FIRST MODE FREQUENCY OF THE BEAMTO WITHIN O.O1S ACCURACY USING A TWELVE BUNOLE IDEALIZATION
BEAM LENGTH = 11 mMASS = 1100 kgElg = 5*104 N-m 2
STRUCTURAL DAMPING FACTOR =0.1
PHASE SHIFTINGFOR THIRD MODE
NOTICE THE ABSENCE OF THE SECOND NATURAL FREQUENCY
8 16 24
SECOND NATURALFREQUENCY
32 40 48 56 64 72 80 68 96 104 112120 128 136
FIRST NATURALFREQUENCY
FREQUENCY (x 10"2 Hz)
FIG. 4 3 : AUTO-RECEPTANCE AT THE MID-POINT OF A UNIFORM SIMPLY SUPPORTED
BEAM.
-180-
AVERAGEDP.S.D. FORCE 5
(NVHz).
KEY TO AUTO-P.S.D. FORCE CROSS-P.S.O. FORCEL FORCE SPECTRA
T • •- F,
SPRING ASSEMBLY
TEN —FUEL
BUNDLES
FREQUENCY (Hz)
«- CHANNEL
-SHIELDPLUG
12
CORREUTED 0FORCES
I MAG
UNCORREUTED F g
' FORCES
CORRELATED Q12 f FORCES
OUTLET
0 2 4 6 8 l O f
FIG. 44: FLUID FORCES ACTING ON AN INVERTED CANDU-BLW REACTOR FUEL ASSEMBLY.
AUTO P.S.
t
i
0
2
0
3
0
*n
5
6
0,
7
0'
8
0
9
0,
10
o,11
12
0
u\.2
V2
\ ,2
\ ~2
2
2
\
2
\2
1 2 r
D. RESPONSE
A4 8
4 6
4 6
A4 6
4 B
4 6
,*.
4 6
A4 6
A4 6
4 8
4 A 6A4 | 6
B
8
8
8
8
B
8
8
8
8
8
4 ,
Î8
lÔ'f
10 f
10 f
10f
10 f
10 f
10 f
10 f
10 f
10 f
10 f
F
lOf
R.fi.S. DISPLACEMENT AT BUNDLE END(mm)
f 0.1 0.2 0.3 0,4 0.5
SHIELD PLUG
SPRING ASSEMBLY
AUTO 0 . 1 0P . S . D .
RESPONSE ° 0 5
KEY TORESPONSE SPECTRA
C 2 4 6F ' F Z F 3 F4 Fs FREOUENCY ( H z )
(NATURAL FREQUENCIES OF ASSEMBLY)
FIG. 4 5 : RESPONSE OF CANDU-ULTF REACT'II; FUEL ASSEMBLY TO THE FLUID FORCES
D E S C K L ;K;I ,:A ;•• • < . 4 4 .
- 1 8 2 -
P I N N E D - P T T N E D BEAM
DYNAMIC MODELLING PARAMETERS
BundleNumber
1
2
3
4
5
6
7
8
9
10
11
12
Length(m)
0 . 5
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0 .5
Mass(Kg)
50.0
100.0
100.0
100.0
IQQ.O
100.0
100.0
100.0
100.0
100.0
100.0
50.0
NELDiara(m)
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
KCST
(N.m)
0 . 0
5x10"
5x10"
5x10*
5x10"
5x10"
5x10"
5x10"
5x10"
5x10"
5x10"
5x10"
(N.m)
5xlO6
5xlO 6
5xlO6
5xlO 6
5x10 b
5 x l 0 6
5xlO6
5xlO6
5 x l 0 6
5xlO6
5xlO6
5xlO6
KENDO
GCST
GPAR
0 .0
0 .0
0 .0
0 .0
Q.O
0 .0
0.0
0.0
0.0
0 .0
0 .0
0 .0
«.m)
oo
iiXp<o(JO
od
II
gwa
^^CM
wB
O 00o o
II II
>• o
VM(Kgs)RH0(kg.m3)U,CD(m/s)CF, CB,FEF
0 . 0
0 . 0
0 .0
0 .0
0 .0
0 . 0
0.0
0.0
0.0
0 . 0
0 .0
0 .0
DIAMH(m)
DEQ(m)
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
' . 0
CALFOR I N P U T DATA
BUNDLENUMBER
123456789
101112
DISMAT (DISPLACEMENT MATRIX,
-0.399390-0.126 758-0.217644-0.299784-0.357437-0.378207-0.357437-0.299784-0.217644-0.1P6758-0.399390-0.0
XX
X
X
X
X
X
X
X
X
X
METRES)
10""1 0 " 3
10" 3
1 0 " 3
1 0 " 3
10'3
10"3
10"3
10"3
10"3
10""
FORCING FREQUENCY
= 1 .0 Hz
FIGURE 46: Typical Input Data used for the Calculation of the
Resolved Forces Acting on a Structure.
(f
(I
.çnn/1447
.5C009?o#4t»O3Afr4.500 0^01,40"9fcAQ.Kpnno?n
.5000447
T = 1 "-' " F•>F p I'i iT (J
. 0 -
. - . 4WiGr
. . . t . 4 3 74
. - . 111071
. .4 t?7»
, . . ? i « o î
. - .40114
FOUI *
o ? r
UF7sF• J y v
7 > F
CE AT
-10)
-10)- I 'M-11)
-10)- ] n j
-1?)
j n
-_t1
f
f
f
I
f
t
t
t -rTt-* ' i
,4Q4P71
.4QOQ7I
F-nii (0. ) ( .4999H1 .-.1737974F-09).1717974F-09) ( .499B714 , 0. )
.oitii7ooF-ir.) ( .499Q747
F_^l 1 )F-i n )
)79=;T;iF-n9)
,-.1( .499B714( .4999611( .4999570 .-.1P4539f.F-10)
,49997m. .54M41BF-1Q)
F-1 n ).-.P4?r>l P« .F- IO) | ,snn]n^7 ,-.oo• n . ) f ,4QO9P*.^ . - . 7 e -, .7114=1 I F . I l l ) ( .SlUHTm , n. ) ( .49Q9B4F. . j OQ*BB4 I f - Ï 0 )
_tT*2*i3£°JlF=iOJ. i. ,4QCl9rt4<- ...OO<.BW41F-1I') f .409B91? , 0^ J. ,5nnCB^jF. l1) | .«0010^7 . - . 7P?77P4F- ln i ( ,50n01B3• . ? 74 Î 7^BF-/WJ / ,4090^47 , , 4MBC.457F-) fl ) | .40OBB31, . . 1 0 P - j * 7 i c - i n j ( .=,(100^,^4 , - .P t3"V-43E-10) I .4999710 , . 14305B9E-10 ). .1 )*.7«"1> - n j ) , , s n « K M . ,4l4?3?hF-l 1) ( .4Q99*,f,9 . . ) 4] 1 1 0<iE-09)
.Ç0OO0R4
.5001*15
. .5417414F-10I.5713A61F-11)
• 0.
.50009*)?•5noo9?o
ftfrl70F-111.?93B444F-10>.14B4974F-n9).1425580F-10).14ln9??F-n9)
!! » .249J0Z F-101
.50OO0B4
.5001431
.5001097
.5001435
.5000962( .500H9?!)
( - - - [ .4900*11( .4OOQ714
ZtblrXnx 1 .4"! 1
( .499QM)t .«inoO447( .4999S70( .K|)009?n
I .401 .40 . . . 1 4PC 1 1QF . QCj) £_ f 4 09 07 1 0
,o=,39";4iF-ir') i .
( .4OQQ714 ( .499QS11 . ,13«?314F-n9) ( ,4999<i70.-.1117inHE-09). .7477VH1E-11)
F-1 1 )
.1501500F-09)-.P93FI444E-10)-.5OB5B?1F"-11).70?77?4F-10)
r. P9410 75F-Î (LI0. ).1191?09F-H9)
-.1512686F-10).1117108E-09)
FIG. 47: CROSS INTENSITY SPECTRUM OF FORCES RESOLVED AT THE BUNDLE ENDS (N2). E; CH COLUMN
OF THE SQUARE MATRIX IS PRINTED FROM LEFT TO RIGHT ACROSS THE PAGE.
- 1 8 4 -
C c i ; f f - T P ( " ' A.T Pun
f l . it n o o<i 1 . _ . i n P 4 C i o o c . . i i )
r " I I 1 , ! ' | F P
( 1
' .
( i
. o o. o n <• Q
F T T f " f ) P f >
OU.-1P7T4
HTCrnrTF FDiirr
•:
' • • n M ' ' "
: t ( Tiv i • • r
. ,?H4n
^TTflif. 1
. _ . 1 A 7POT
•>•• f T F - M ' - .1
T
•-^
T
M
T
r
' -
K
' -
F
0 4
']>•)
' ! . .
M i l
( i f
( C
O i l
l . - l |
' • "
• I
M
M
l |
' !
M
tr -<
f- ,
I •••
Fnarf ' : ; : . i n i - i i » .• T (•'•'••! ' i t . ' I • 11 «-
f 1 . 0 0 0 0 7 » . . , ' , i ?n,-; ' . . l i- - 1 :•• )
F',;-iri" •;"• F r i I I ' • ' " " t ' ' ' ' : ' t U ! I ! l | t
J I ? . _ . 1 h ~îf••-••• •>> - -• - )
I i F' .crt- ' - i f I - T C n - / T ^^•^.•• r,r -a i • -.. >| c
: ( 1 . ' i f " 1 i . i . - , - > - • 1 7 / , 1 - i - 1 • • ' )
! nicfoftr nwi'. i . i . - ' i r ' •• T k-.. , , i : , . , - . •. | [.
!
j ( , ' i i i ' - . W U . . 1 ..< •> . . . • , • ' « ' - ' I - ' . ' )
, ' I T c r n c - T i , • : , , • ( - , • '•'. • r j i M . T » • . •'> . . i r | . ] .•
; ( 1 . f l - . W r C, . - . 1 1 . • . , • . - ! • •. k- _ ' , . )
! I I C f n r T i - ! • • • ! ! > . • • • • . . . . - . - T i I i ' • * • • , ' i t :". | f \\ i " I I
j ( 1 . f . r . . - . - , , 1 . . 1 -.<. s k 1 i . -_ . ( . , )
I . M t r n r n k • . . . - - • «.. • i T i i . • • -.7 i ' . i ; . • • i i-11 • , I c 1 ;
FIG. 4 8 : DISCRETE FORCE SPECTRA. AMPLITUDES AND RELATIVE PHASES, GIVEN BY COMPLEX
NUMBERS, OF THE FORCES ACTING AT THE BUNDLE ENDS FOR A SINGLE SPECTRAL LINE.
anTO PnwFV ^P
( #snrt040f>
OF rnwCF ÛT FNn ftF
n.
AUTO POWFP qPFTTf lM OF FODTF AT f^Tt OF RlifctOI. F ?
AUTO POWFO CPFCTPIIM PF FflOfF 4T r " n OF OMM I F ?
UMlFflPM «F
.1,00040*
.4PPH714,^00141=1.4QOPO1?.snpo7f.n
r .soni4m.491K73*.S0004RQ.SOD040f
( o.
. n.
. n.
• 0.• o.• o.
. o.
. .0.
• n.. n .
i
)))]
:
i
OF FPUCF AT Fnn >OF UMM'H F p
. )
DF FinCF AT F'in nP onN'tl F o
AUTO AT Fnn -IF PMM-H.F ] 0
MM OF FPDfF AT Fvn I F mi U
AT F'Jr I F HliMll| F \f
i n . n.
BOOT MF«M
.7i)7]in
.7070171,7fl7?nB?.7070 •>!?.7071*IS,707011?.707P0B?.707017-3.7071413.70711"0.
• 0,• o.. n.• n.• 0.. n., 0,« o .• n.• 0.. o.
)))))1)
)
1))
FIG. 49: SOME TYPICAL OUTPUT
FROM SUBROUTINE CALFOR.
-186-
APPENDIX I
DVNMOD Listing
c c c û û c c o o o c o c o Q c o o o c o c c e c o o O û C Q C û c c c c c c c e c c c c o o c o o o e c o o o cC C C C C C C O C C O O C C C C C C C C C C C C C C e O C O O ' 7 ' C C C C C O C C O O C C C C C C C C O C C C O Q ©> > - 5 - > V ? - > - > > - > - > > > . > > > . > > > . > > . > V V > - > > > > V > > > > V > - > . > - > > - > > - > > - > > > > > - > > - > V > - > > - >CCCCCODQaoCCOCCOÛOOOOOOOCOOCOOCCOOCûOÛOCCCCiÛCCCOÛCQCiOOQÛO
o o i- a •- >U 11. ZU. 2 = u O<j o ?iu 4 a c c a <r u- 4 •
— <J •»© J l U • - 2 < X lu U"C D > K « - I - K Zc z -J «J < ir cU C Z O K C • -
i- 2 lu u. e • z uD < a c c ir « zC " û r c i i . U.-Z 3x » u. u- u. i c » a t - u. UJ»- If) _) X lu Z _J lu Z X»- lu ce >- u••> tr. a. »- UJ a »-2 0 4 z o i- a »-• lu
a ^J IJL 4 UJ 4 CÛ cf' U- aao»-« o a > - \fï z •— -~ v ~ cc u. _i »-<E • c r » - « < i * z « b .
x o a a c o UJ Z ? 2 U , K u a c v :I -UJ< 3 Z UJ > *- >- <. z u H- a. a«-••-» » - 4 f - UJ C 4 C 2 U. a U U.3 JIL 4 < _ l Z a '-' © O U- UJ I-aa z UJ _J UJ •-« t - c a L .cau
c a u ir — u. w. z z. zu.4 I Ui 2 U O 2 4 U . C U . 2 U J 4
_J «T UJ V> U 2 Z? • • •; z in U J U ) >• 4 3 j J jiu 2 j»ru a. 4 (j a UJ o U . C C C 4
4 K a C >- tT. U. C- UJ UJ UJ 3
UJ C C CL »- <X 4 hHtfUli.Vii4irju- z i u r • c c c x i r oCt O 4 .JU1U. Zi • C UJ • UJ UJ U.' O U) O O O C
z ji/ ica a • z a v> a ne ct- inh-ui-z< > o j,4ibh UJ» »- a u. a a i - a cat—a;
O K Z - J O < » »-UJ O V U If «If l S U C Q U . t t U . l -W ^ ^ * 4U:U.'J4 \f C. JU.3U.l(/'IUlLri^|x.4lL'CliJ4u"2z a r _ i K zr ac .«JC 'acc"—a x a u. a cZ « - * C C U. 3 U1 < U _l » Q » U . * C • C • •c 4 c e x c i u a h < i h t - h r : i - u . c ( - O H <O I T J tt<2(- U . 2 C b C k U U k . C J l L a t L U . l i U Q
u: u* > • z «4 a*~ *i « * 2* ? J « wU4U. x u z. c V' d. K et »-< i - H « M L a t - c i - a
•J j u : z u- K- a a c c i ^ c C U - J C c a c '
O J C J •^ -at/'*-> in- 4 is c v tx ( r e UJ*"" m Z J U Z (t Ul 4» lliîtt2^2CU)2tt 2 h* Z (tUZU.' ' _J4CU*a U-G < C h - C _ i C W. C U. UJ O 4 O O«? ** J 4 IJU }4 U-' ^ l^ ^* U. 4 ^ "^ r ^^ Cï ^^ 1 1a c u j Z f a o u j a a 4 z*~ • c a •- UJ 4 c «-• • • • • « c • •tfjtu. u. u tr. Q 3 u. o —< iv rr •* -J i ra <5 K
c 2 a •- <s o c u. trj h . « c x z a bv r tuu. o u u. K u u. a u. a
*
s
u.
*1+IIcz0a
"JJ1
_ac
•4
<f
a
cHC
DCU,
•
tâ2uu*-
a
•0
a»
a
c
ar
C0
z2cza(Scaa
uucuuu<J
u
u0
( Juuu0uuuuuuu0
uuu(Ju0<J
uuuuuuuuuuu( Juuu
*
0
>
«»*
•
acc-aa.
- J
luOC
<J
S
z
<
a
V.
cx «C J
5 22
1 1 . X<J2 3U. O
^2 U.l i j0. _lU. lu
_J
a ac <t- a
UJ
u a^ u.1 y.
O «X- J
0 u.
0 u.cs az c>- r -c u
4u.a
<_ J i r .
— ^.b - C
u. u.c •-lu
Z Xc 1-a1/ c.a; 2a <Z If
^ tj
c 0c u.lu 0
** z1- Czui a
H X •
W H UJ
C UJ a™ *"" C'CITILu. 2a 0 u.a a ir
1 x« 0 1-a Kc 0O !/• Ka u.1a i-i UJ
-1 wu. a zX S C1- u. a
in vtV, lu
z 2 aU "t
tu i/ tt•- i U.1- 0 >• - • - • c_J H-
x z£lilj<r. a Di r . u - <o u . »air. lu
u zZ H- 4
UJ CC IT Zw X< h 2
CU; • «-"
•- a <101- a.UJ U C•- : u_ J I - >-
Û lu_ u. «< X 4u n0 u.ir oc ._l Z<a V- «
UJ
c z cI < 32 H ->• a *•r i. t
u au. 2.0 <
•
UJ
uzUJ
cUJab .
c a.1 cz0 a
•-CC l ua aa «A cU.'
Î3 S^Ê•> u . t -c a «a if UJ
a » a0 a
u cX <A U
4
« c«-(J UJ
1/1 V luimi CL V"
a a **0 x mU. UJ Z
S UJ CC (LC _l
< « a(t U H
CCl/Ht- u. a
u ir
u- c aZ U-lilC 3H C C
v. u. a.
a UJUJ a uX 4 4»- a
• »- 4
t
saci t
u.K-
UJ
aac
74/74 OPT=1 ROMND=*-*/ FTN 4.6*433 7R-09-1B 15.01,58 PAGE
70
90
ion
110
CALCULATION PF RANDOM FOPCEO RESPONSE.(TWO OATA SFTS.REFEP TO SUBROUTINE RANDOM.;
fl. LORIC PAO«iMFTFRc. PECEPTANCE AMD PFSPON-iE SPFCTRA FOP THFCALCULAT I ON OF «FSOLVEO APPLIFP FORCES.
(THREE DATA SETS,RFFFP TO SUBROUTINE
INPUT...
O*T* SET 1......DFPUGGING AND PLOTTING PAPAMFTERS.
DP-UG x 0 ...NO 0FB|if,RIN(5 INFORMATION RFOIIIPFD.= 1 ...EXTRA INFORMATION PPINTFO FOP DFBMGGING.
PLOT = n ...MO GRAPH PLOT SUPPLIFO IN THE OUTPUT.= I ...MpHESHAPES AND COMPlFx FPFOUENCY PLOTS
ARE SUPPLIED WITH THE OUTPUT.PWO x ...BUNDLE WIDTH RATIO.SOP = ...*<OOE PLOT DEFLECTION RATIO.XL = ...LENGTH OF X AXIS (INCHES).VL = ...LENGTH OF Y AXIS (INCHFS).
NPEPFP. s ...NIIMPFR OF M00ESHAPE5 PER PLOT-FRAME.
OAT» SET ?......«STRUCTURAL PARAMFTFRS.
= 0x 1=-1
GL(N)
DIAM(N)KCST (N)KENO(M)KPAR(N)GCST(N)GEND(N)GPAR(N)DCTOPX
OFTOP
...RPAvITATIlN EFFECTS IGNORFD.
...FUEL ASSFM&LY HANGS OFLOW ITS SUPPpRT.
...FUEL ASSE^PLY STANDS AMOVE ITS SUPPORT.
.ACCELERATION HUE TO GRAVITY ("ETRE«J/cEC»»?)•VFCTOP OF HUNDLF LENGTHS («ETRES)..VECTOR PF PUNOLE MASSES (KGS.l..MirMRFRS OF ELEMENTS IN FACH R-IINnLF..DIAMFTERS OF FLEMENTS IN FACH BUNDLE (M.).
CCCCCCCCCCCCC-Ccccccccccccccccccc
.CENTRAL SUPPORT TUKE STIFFNFSSFS (N.WETPFÇ) , C
.ENDPLATF STIFFNFSSFS ( W F U T O N . M F T R E S ) .BUNDLE PAPALLELOGRAMMING STIFFNFSSFC (N.M.)
C.S.T. STRUCTURAI. DAMPING FACTORS,E FNDPLATE STRUCTURAL DAMPING FACTNPS.
"UNDLE PARALLELfKJPAMMING S.D. FACTOpe.FXTPA STRUCTURAL DAMPING FACTOR ASSOCIATEDWITH C.S.T. CONSTRAINT AT THF SUPPORT.
EXTPA STRUCTURAL DA M P J N C FACTOR ASSOCIATEDWITH ENDPLATE CONSTRAINT AT THE SUPPORT.
C0CFCB(Ni3)
DIAMH(N)V«(N)
OEO(N)FF.F(N)
CCcccccccccc
...FLUin DFNSITY NEAP EACH R.HNDLE (KGS./M.»»3). C
...ELOh VELOCITY AT EACH BflWCF (M./SEr.|. C
...COEFFICIENTS OF DPAG (C D . M . / S F C . ) AND CFRICTION (CF). ASSOCIATED WITH EACH RUNOLEf C
DATA SET 1......FLUID PARAMFTFRS.
PHO<N)
AND RASE OPAG COEFFICIENT (CR)..,.HYDRAULIC DIAMETER AT EACH PIINDLE (M.)....HYORODYNAMIC MASS OF EACH BUNDLE (KPS.)....EOUlVALENT DIAMETER OF THE FREE FND (M.)....FREE END FACTOR ASSOCIATED WITH THE SHAPE
OF THE FREE END OF THE FUEL ASSEMBLY.
(THE INTEGERS IN PARENTHESIS AFTEP EACH VARIABLE GIVF
0YN"OOOYNWODDYNfODDYNM0DDYNMOODYNMODDYNMODDYNMODDYNMOODYNMODDYNMOODYtJMODDYM'OOOYNWOODYNMOODYNMODDYNMOO0YNM00DYNMOODYNMODDYNMODOYNMODDYNMODOYNMODDYNMOODYNMOODYNMOODYNMODOYNMOODYNMODDYNMODDYNMODOYNMODOYNMOODYNMOOOYNMOODYNMODOYNMOODYNMOO0YWM00DYMMOODYNMODDYNMODOYNMOODYNMOODYNMOODYNMOOOYNMODDYNMODOYNMODDYNMOODYNMOOOYNMODDYNMODOYNMOOOYNMOOOYNMOD
60616?63646566676469707172737*757677787900fll«2B3f»405A6fi7RS«990919?93949S96979B99
ino101102103104ins106107109109110111112113114115
74/74 0PT=1 ROIIND=*-»/ FTN 4.6*433
115
130
135
1*0
155
lf.0
C THF ORDFP (IF TWF M/STPÏX.) C
c cc cccccccccccccccrccccccccccccccccccccccccccrcccccccccccccc.rccrcccccccccccC LAMPS FUNCTIONS.
cINTEOFR AOr>,Ar>m,ASsIGM.A<:SIGNC,O.DELFTE,DIV.GFTCn:. .GFTROw.ONE.
. ONERLO.ONF.HOV.OUTPUTCOIITPUTL.OUTPUTT.PAGF.POUFRN. SCALAR,, SCALAPC.SDIARHI ,S0IAGL0,SI6NMOD,SMOLT.SMULT3.SPECIAL.SUft,. S I ITOl .SIIHMAT.SURROW,TRANS.TPIÎNV.7FROCOMPLFX nFT.FLFMF.K'ï
LAMPS STORAGF ARFAS.
I N T F G F P r > I R ( l ? O O I , R n W ( i a O O > . C O L ( 1 ? 0 0 ) « L I N K ( 1 2 0 0 )COMPLEX SPACE (330(10)LOGICAL LASTCOMMON /M?/0IP/M3/P0W/MA/CnL/M5/L!NK/MA/SP«CF.COMMON /M1O/LA<5T/M<Ï9/COMCAPD(P)
MATRIX POINTERS, FUNCTIONS AND VARIABLES.
INTEGER A,AnPnx.ALPH»,ALPHA?3,ANGLE.APFA,ATFNn,B,BFTA,BOlMV,F>WR,C,CR.rn,rnrFCP,CF.C»"AT,0BliG,DBY0X.nCTnpx,DEFLFC,nETOPX,OFO.OIAM,niAMH,OKPARD,FA,EB.FP.FEF,FL,FPEO.G.GAMMA,fiCST,GENn,GPAQ,H,HALF,HPHOND.PHIS,PI,PLOT,OE0,OFl,nMn,OHl,070,P.PHO.snCBDT.pnCSTX.SDEBOT.SnENOX.SDP.T.TCMAT.TFO.TFl,Tf?.TPOT.TTRANS,TwriPI,U,Vll,Vl?.V?l,V2?,VCSTIVFFF,veND,VG.WM,VPAB11.VPAR1?,VPAR21,VPAR??,VR.W.XL.Y•YL
LOGICAL BOTPIN.nFBIIG.LSING.MIDPIN.PLOTS
DFFINF STATEMENT FUNCTIONS,
OBYDX (K)=MULT(fiAMWA, r>I V( MULT ( TRANS (BFTA),K) , MULT (NFG (KAPPA) ,L))>ATFNO(K)=SURROU(Ann(K,SMULT3(HALF,L,0BYnX(K))),N)
DIMFNSION ANO HIDIP, AND DEFINE VALUES OF CONSTANTS.
CALLHALF=ASSIGNJSCALAR(0.5.0.0>)TW0PIiASSIGS((<5CAI.AP ( 6 . 2 8 3 1 8 5 3 , 0 . 0 ) )P I = A S S I R N ( S C A L A R ! 3 . 1 4 1 5 9 2 6 5 . 0 . 0 ) )
SET DEFAULT VAl.tlFS.
0BIIG*PL0T»7l'P0(l,l)*ZERO (1,1)
CnCFCB*7ER0j3,i)
1T0
C INPUT DATA.C
ï CONTINUECC DATA SET 1......DEBUGGING AND PLOTTING P4BAMFTFRS.C
7R_n9-li
DYNMOODYNMODDYNMOOnyK'MnnDYN^'ODDYNMODDYNMOODYNMODOYNMOD
nYNwnoDYNMODDYNMOODYNMODDYNI'OOOVN'MODDYNMODDYNMODDYNMODDYNMODDYNMODOYNMODOYNMODOYNMODDYNMODDYNMODDYNMODDYNMODDYNMODDYN"ODDYNMODDYNMODDYNMODDYNMODDYNMODOYNMODDYNMODDYNMOODYNMODOYNMODDYNMODOYNMODDYMMODDYNMODDYNMOODYNMODDYNMODDYNMODOYNMOODYNMODDYNMODDYNMODDYNMODDYNMODOYNMODDYNMODDYNMOD
15.ni.58
1 1»-1)7lia119l?01?11?21?31?4
1761?71?81?913013113?133134135136137139139'14014114?143144145146147148149150151152153154155156157lSft
159160161162163164165166167168169170171172
P» E
00ooI
PROGRAM DYNMOD 74/74 OPT=1 FTN 4.fc»433 78-09-18 15.fll.SA PAGE
175
iflO
IPO
195
CALL READ7 (4HDRIIG.4HPL0T.1HPUP,
?HXL.?HYL,
DRUG,PLOT,PWP,SOR,XL.YL.
IF(LAST)STOPftHNPERFR, NPFPFR)
CALL REAimUHY.. 1 MR ,
1"L.. 1HM.. 3HNFL,. 4H0IAM,
4HKCST.. 4HKFMO,. 4HKPAR,
iHGCST.. 4MGFN0.. 4HC-PAB., AHDCTOPX,. fiHPFTOPX,
Y,
c-.L,»,NEL.DIAM,KCST,KEND.«PAR.GCST,PEND,FPAR,DCTOPX,DETOPX)
SHPIA"",
CALL READ7 MH9M0. PHO,1HII, U,
CDCFCB,DIAMH,VM.PEO,
1HFFF, FEF)NsGETPOMIL)IF(COL(CPCFCR).EO.l)
COCFCB=ASSir,N(OMTPUTL (14HASSUME0 CDCFCB,?.TRANSIOELETF (CDCFCB) ) ) )CO=ASSIGN(SlincOL(COCFCP.l>)
(CDCFCR.21)
PLOTS=0,n.NF.oF»L(ELEMENT(PLOT,1,1))
CALCULATF THF COOPOTNATF TRANSFORM MATRICES,
CMAT=ASSTGNC|MULT(LOWFP(N),D<L))))
C CALCULATE THE COORDINATE PEDUCTION MATRIX R,IF THE B1INPLE STRING IS ENO-PINNEO,MID-PINNED.00 BOTH.R IS USED TO REDUCE THE NUMBER OF DEGREES OF FRFEDOM OFTHE VIBRATING SYSTrM,
CALL PINNED(JP.L,N.R,B0TPIN,MIDPIN,KCPOT,KER0T.SDCHOT,S0EROT>
cccccccccccccccccceccccccccccccccccccccccccccccccccccccccccc.cccccccccc
HYNMOOOYNMOOOYNMODOYMMOOOYKiMOODYNMODOYNMODDYNMOO0YNM0ODYMMODOYNMODDYNMODDYNMODDYNMOOOYNMOOOYNMOOOYNMODOYNMOOOYNMODOYNMODDYNMODDYNMOOOYNMOOOYN»OODYNMODDYNMOODYN"OODYNMODDYNMOODYNMOODYNMODDYNMOODYNMOODYIJMOOOYNMODDYK'MODDYNMODDYNMODOYNMOOOYMMODOYNMODDYNMOODYNMODDYNMOODYNMODDYNMOODYNMODOYNMOODYNMODDYNMODDYNMODDYNMODDYNMOODYNMOOOYNMODDYNMOODYMMOD
173174175176177178179ion
1«31»4JP51R61P71»81B919019119219319419S1»619719R199?no
?04?05
?11
I\—»O5
I
PROGPAM DYNMOn 74/74 FTN 7 B - 0 9 - 1 B 1 6 . 0 1 . 5 8 PAGE
? * 0
?T0
pan
c cC STRUCTURAL rrjrPRv I F » » ^ , CC CccccccccccccccccccccccccccccccccrccccccccccccccccccccccccccrcccccccccccC CALCULATE THF KINFTTC FNFPGY CnFFFICTFMT MATRIX T.C
ALPHA=ASSIGNC. SKI iLT (SCAL«R<n .5 . " .« )
(*> ) 1GAMMA=ASSIGMC(AnO3(TRANS(KAPPA),. AOr>(SPFCIAL<N',N.l.l) .SPECIAL IN.N.N.N-] ) ) ,UFG (SPFC1 AL (N.N.N.N) ) ) )JDIIMMY=OMTPIITT(11HATENO(RHO) .ATFND(RHO))»
. OUTPUTL( l lHATENn(VM/ t ) .? .ATEND(01V(VM.L ) ) )JOUMMY=OIITPIITT(BHATFMO(U) .ATENO(II) )*OUTPUTT(BHOBYPX(U> .OBYOXCUMTTRANS=MIILT1(TPANS( ALPHA) . 0 ( M ) , ALPHA)TPnT=0(SMtlLT3(SCAL«R(O.OP3333333.o.O) .M.poiiERN ( L . ? ) ) )T=ASSIGNC(AOO(TTRANC,TPOT))
CALCULATE THE POTENTIAL ENERGY COEFFICIENT MATRIX VEFF.
VG=ASSIGM (0 (MtiLT (MIPUT (Y ,G) ,M<iLT (TRANS(ALPHA) .M) ) ) jVCST=ASSIRNr(MiiLT3(TPANS(PFTA) .D(KCST) .PETA) )VFND=ASSIGNC(M|JLT3(TPANS(PFTA) ,D(KEND) .BETA) )
INTRODIiCF 400ITIONAL STIFFNESS TERMS I F THF FREF END OF THE PtINPi ESTRING I
. ( A O D ( n e L E T E ( V C S T ) . J O I N V ( 7 F P 0 ( N - l , N ) , J O I N H ( 7 E R 0 ( 1 , N - 1 ) . K C P O T ) ) ) )IF(ROTPIH)VFNO=ASSIGNC
. ( A O n ( P F L F T E ( V F M O ) . J O I N V ( 7 F R O ( N - 1 , N ) , J O I N H ( 7 F P O ( 1 . N - l ) . K F P O T ) ) ) )VPAR]! '
INTRODUCE STRUCTURAL PAMPINGF1R5T FORM THF MATRICES REOIRFD TO DFSCPIRE THF FXTRA STRDCTURAL
AT THE E M P CONSTRAINTS OF THE P.UNDLF STRING.
SDCSTX = J<UNl ( J O I M H ( S M I I L T 3 (SCALAR(O.0 .1 .0 ) .HCTOPX.SllPROW (KCST. 1 ) ) t. 7 F R O ( 1 . M - 1 ) ) , 7 F R O ( N - 1 , N ) )
t.OELFTE(SOCPOT)))
) , 7 F R O ( N - 1 , N ) )<;<;iGNC<AnO(PELF.TE(SDENDx) .DFLFTF rSDFPOT) ) )
= 4SSIGN(CMiiLT (SCALAR ( 0 . ( 1 , 1 . 0 ) .GCST) )IMGFN0=ASST'5NfSM!iLT(SCALAP{0.0.1.0) .GENO) )PKPARO* ASS I «N(D(SMIILT3( SCALAR ( 0 . 0 . 1 . 0 ) .GPAR.KPAR) ) )Vl l=ASSIGNC(Anni(VG.vCST,ADD3(VPAPll ,SM()LT( IMGCST,VCST).ADD
Vl?=V?l»ASSIGMC(AnO(VPARl?,NEG(DKPARD)1)V??*ASSIGNC(*.r>D3(VEND.VPAR?2,ADD3(SMULT(IMGEND.VEND) .SPENOX,
DKPARO)))VR*ASSIGN(NÇQ (MIJLT (TRI INV(V22 ,1 .0E-1S.LSING.NR7.J7ERO) .V?1) ) )IF(LSING)PRINT 91 . ( INT (PEAL(ELFMENT(J7EP0 .J . l ) ) ) , J *1 .NR7)
PI FORMAT!» V2? SINGULAR — REDUCED DEGREES OF FREEDOM»/ T10 .
DY'JMlin
PYM«OOOYNMnP
OYNMOODYNMODDYN«OOOYNMOODYNMOOVi TJMOl»
OYNMODOYNMODOYNMOD0YNMO0OYNMOOOYNMOOOYNMOODYNMODDYNMOOOYNMOOOYNMOOOYNMODOYNMODOYNMODDYNMODOYNMODDYNMOODYNMODDYNMODOYNMOOOYNMODOYNMOOOYNMODDYNMOODYN"ODOY'JMOOOYNMOOOYNMODOYNMOOOYMMODOYNMOODYNMOODYNMOODYNMOOOYNMODOYNMOOOYNMOODYNMODOYNMOOOYNMOODYNMOD
?40?41?42?43?4*
?*S?4A?47?4«?*9?S0?si?S2??3?«;*?S5
?57?5fl?59?60?ftl?ft2?63?A4?(i57>f-b?f-l?AH?A9?70?71?72?7327*?75?76?77?78?79?»0?«1?B2?B3784285?86
OI
PftOGRAM DYNMOD 74/74 nPT=) FTN
300
301
310
315
3?fl
330
335
340
« D<;i« j.T)s»S! (J) =AP«ITRAOY»/» AN' ira "ILL BE VALID PROVIDED THAT»/ T10.» KPA&U) = 0»/» FO« J =•
= OIITPIITT|BMDFT(V??) ,SCALARC<nET(V??) ) ) «OIITPUTT (PHVP.VR)VFFF=ASSIGNC(AOD|VI1 ,Ml>LT ( V] 2 . VB) >)
Ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccrccccc cC HYPROOYNAMTC TFOMS. Cc ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccC CALCULATE THE ENFOGY AND DAMPING COEFFICIFMT MATRICES TFo,TFl,TF2.
cTFn=ASSI«NC(NFR(O(SMULT3(VM.U.I.I) ) i)TFl=ASSIf!NC (SUR (MULT (TRANS (ALPMA) ,
. n(SMiiLT(VM.II) ) ) .MULT(P(SM|)LT(VM,U) ) «ALPHA) ) )TF?=AÇSTGNC (Ann (MIILT3(TPANS (ALPHA) ,
. D(VM),ALPHA).n|SMULT3(SCALAR(0.083333333,0.0),VH,SMULT(L,L>))))CC OPTI"I7E CODf ?I7F AND SAVE ENERGY MATRICES.C
CALL nELFTFx(T.fl)T=ASSIGN(T)
TFn=A«;SlGN<TFft)TF1=A<?«!TGN(TF1)TF2=ASSTGN(TF?)CALL CLFANIIPCALL SHOWMFM
EVALUATE THF H Y D R O U Y H A M I C MOMENTS.
MOfFNTS OFn AND OF1 CAUSEfl PY FIIFL ARSFMRLY FNO FORCFS.
Ç.O.O) . S M U L T 3 ( A T F N D I B H O ) .DEO.DFO) ,.ATFND(II) ) )
Fn=NEP(SM|iLTT(<:im(ONF (1,1) .FFF) .ATENDIPIV (VM.L) ) ,ATEND(U) ))f3F0=ASÇTGN(Si|B(«IILT[r)(«;MULT(L«A0D(EA,SMnLT (EB,ATFND(U) ) ) ) ) ,,J0INH(7FO0(N.M-l).ONE(N.I))),O(SMULT(FA.L))))OF l=A5STr,MC(SMULT (FR. MULT (L.TRANS (L>> ) )
MOMFMTS OHO «NO OH1 CA|j<;FD BY NORMAL COMPONENTS OF VISCOUS FORCES.
SMULT (HALF.SMllLT3(PH0.NEL,niAM) ) )H=ASSIG^I(SM|ILT(HRHOND.ADD(CD.SMIILT(U.CF) ) ))
. [MULT ( Ann (O*)FPLrt(w> .SMULT( SCALAR!?. 0 / 3 . 0 , 0 . 0 ) . I&FNTiN) ) ) , 0 ( D ) >OH0 = ASSTRNC(MllLT(n(NEO(L) ) «ADD (D (SMULT3 (HALF. L,FL) ) •
. M|ILT(OMEROV(N) >n(c!MULT(L,FL) ) ) ) ) )OHl=ASSIGNC(MiJLT(n<NEG(L> ) «ADO (KULT (fi (SMULT3 (HALF,H,L ) > ,ALPHA?3) ,. MULT3(OMEP.OV(N) ,n(SMULT(H,L) ),ALPHA) )) )
T8-09-1A
PYN«ODDYNMOODYNMODDYNMO0DYNMODOYNMOD^\ ^ Ék | àj A ^\
i)T N MOU
DYNMODOYNMODDYN«OODYNMQDDYNMOODYNMODDYNMODDYNMODDYNMODDYNMODDYNMODDYNMODOYNMODDYNMODOYNMOOHYNMOODYNMODDYNMODDYNMODDYNMOOOYNMODDYNMOOOYNMOODYNMODDYNMOODYNMOODYNKODDYN'»ODDYNMOODYNMODDYNMOODYNMODOYNMOD0YN>'0DDYNMODOYNMODDYNMODDYN«ODDYNMODDYNMODDYNMODOYNMOODYNMODDYNMODDYNMODDYNMODDYNMOn
I DYNMODDYNMODDYNMOO
15.01.5S
?R7
?B8?B9?<)0?91292
?93294?95?96?97?99?9930030130230330430530630730830931031131231331*3153163173183193?03?13?23?33?43?53?63?73?fl3?933033133233333433533633733B339340341342343
PAGF
I
VD
I
PROGRAM DYNMOD 74/74 FTN 7fi-n<J-lS 15.nl.5(1 PAGE
3*5
350
355
365
370
375
3B0
390
395
C MOMFNT MATRIX 070 CAIIÇÇO RY i.0^1 TuniN.1L VISCOUS ANO PPFS^MRF FORCES.
ci n p n x = A n m ( S " I I L T < Y . S " U L T 3 (<"> , R H O . A P F A ) ) , N F r . ( Ç M I I L T ( K i . ,
. n i V ( D I A M , n i « M H ) ) ) . N E G I S M U L T ( R H O . S M n L T I M A U f A . I L D P Y D X ( > > ) ) ) ) I0 7 0 = A S M «NC (Mill . T ( P ( L ) . A D O ( D ( S U f l ( S M L H T 3 ( H A L F , L . A n p n X ) . " U L T (ONFROV
. ( N ) . S M | IL T ( L ^ S M " ( F L . A D P D X ) ) ) ) ) «MULT (ONFROV ( M) . D (SMI ILT (L t F l . ) ) ) ) ) )CALL DELETEX(T.n)VP=ASSIGM(VR)
cccccccccccccccccccccrcccccccccccccccccccccccccccccccccccccccccccccccccc cC ASSEMBLE THF MATRIX F0II4TI0N OF MOTION, C
c cC A.(r>/nT)»»?."Hi « P.ID/DT) .PHI « C.PHI = n. cc cccccccccccccccccccccrccccccccccccccccccccccccccccccccccccccccccccccccccC COLLFCT TOHFTHER ALL COEFFICIENTS OF (O/DT)••?.PHI.
c
cC COLLECT TOGFTHER ALL COEFFICIENTS OF ID/nT).PHI.
cC COLLFCT TORFTHFP ALL COEFFICIENTS OF PH I .C
C=ASSIGN(SltR(AnOIVEFF,TF0) , AD03 (OEO ,(3H0 ,070) ) ICALL CLEANUPCALL
C
CI F CTPIICTIIRF I S PINMFn.llSE P TO REDUCE OPDFR OF «ATRTCF.S A.P AND C .
= «<;SIfiNC(MllLT3 (TRANS (P) .DELFTE (A) ,R ) )I F l H O T P T M . O R . M i n P I N j H s A S S I f i N C t f l l L T t M T R A N S I R j . D E L F T T l R l . R ) )IF (F )OTPIN .0R.MinPI»J )C = ASSIRNC (MULT3 (TRANS (R) .DFLETF (C) , P ) )IF (BOTPIM.OR,MinP IM)N=PFTCOL(P )I F D E B R j n Y = O I I T P | I T L ( l P H N F W FOUATIONS — A , ? , A )
•OUTP| ITT(1HP,H)»OUTPMTT(1HC,C).SCALARC(nFTIA) ) )
ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc cC CALCULATE THE EIGENVALUES (M||) AND EIGENVECTORS (MOOES). Cc cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
RRINV = A<5"; I RMCl INVERT ( JOINV ( JOINH (7ER0 ( N , N ) , A ) , Jf l lNH ( A.P ) ) ) )EP=ASSK,MC( jn iMV(JOINH(NEG(A) t ? E R O ( N . N ) ) , J 0 I N H ( Z E R 0 ( N . N )WiASSIfiNC(MUl.T3(SCALAR ( 0 . 0 t l . O ) . R R I N V . E P ) )M0DES=AS<!IfiM<M004L(W,MlP) )
CTF(DEPUfi) jni)MMYxOIITPUTT(?HMU,MU)»OUTPUTT(5Hf ODES.MODE*)
CC CALCULATF NAT1IR»L FRFnilENCIES AND MODE SHAPFS.C
3"50
OYN"Oi)DYNMODDYNMOODYNMODOYNMODOYNMODDYNMODOYNMODDYNMOD
DYMMODDYNMODDYMMODOYNMODDYNMODDYNMODDYNMODOYNMOODYNMODDYNMODDYNMODDYNMOnDYNMODDYNMODOYNMODDYNMODOYNMODDYNMOODYNMODDYNMODDYNMOODYNMODDYNMODOYNMODDYNUODDYNMODDYNMODDYNMOOOYNMODDYNMODOYNMODDYNMODDYNMODDYNMODDYNMODDYNMOD
3553563573583593*03 M3»>2363
365366367
36937037137E37337*37537637737fl3793803P13P23R33P43R5in 63R73083«939039139339339*39S396397398399400
PROGRAM 74/74 OPT=] FTN 4.6*433
4 0 »
«in
4 3 5
4 * 0
44S
450
.nfLFTF (PHIS)>
jnilM>iy=nnTPMTr iPnMCIRFpfn FRFo (H7) « P . P A G F ( J O I N H < M H , F R F O ) ) >. •nuTPIITTUO^nFFLFCTrON.OFFLFC)
IF(PLOTS)CALL 4or.sMni>«u)TF(PLnTS) COMr«on(3)=ioH (RFAL)IF(PLOTS) CALL ROAFM(PHIS,«Mr,LF.L.)iL.VL.BWR,<:nn.wPFRFD)CALL CLF»NI|OIF(PLOT<!) rnMc»Rn(.?) = inH (SIGNuori)IF(PLnT«:) C»LL GRAFM (SIfiNMOP (PHIS) «SIGNMOn (ANGLE) ,L ,X| , Y L .
. PWR,«!nP,NPERFP)CBLL CLF4NUR
Ccccccccccccccccccccrccccccccccccccccccccccccccccccccccccccccccccccccccc cC OPTIMIZE CORF nfFODF CALLING THE SUBPOUTIMF PACKAGES. CC Ccccccccccccccccccccccccccccccccccccccccccccccccccccccccrccccccccccccccc
CALL DFLFTFX(ALPHA,0)
CALL CLEANUPCALL SHOWMFM
CALCULATE THF TRANSIENT AND STFADY STATF RESPONSE TOEXTERNALLY APPLIED HARMONIC FORCES.
CALL FORCFS(R<1TPIw.RRINV«CMAT.MinPIN,MOOFS,Mii,N,P.TCMAT|
CALCULATE THF TRANSFER FUNCTIONS RFCFPTANCF AND MOBILITYBFTWFFN POINTS ON THF STPliCTUPF.
CALL PFC»OR(ROTPIN.RRINV,CMAT,JP.MinPIN,MOOES,MU,N,R,TCMAT)
CALCULATE THF RES'OWSF TO RANDOM FORCFS.
CALL RANDOM(ROTPIN,BRINV.CMAT,MIDPIN.MOOFS,MU.N,R,TCMAT)
CALCULATE THE RESOLVED FORCES FROM REEPONSF DATA,
CALL CALFOR(ROTPIN,«RlNV,CMAT,JP,MIDPIN,MODFS,MUiN.R..Tf:MAT>CALL OELFTEX(ALPHA,0)CALL CLEANUPCALL SHOWMEMGO TO 1END
7B-09-1H
DYNMOD!YNWODHYMMnoDYNMOOriYN"O0DYNMODOYNMODOYNMODOYNMOODYNMODOYtJMODOYN»ODDYNMOOOYNMODDYNMODDYMMODDYNMODDYNMOD.DYNMOOOYNMODDYNMODDYNMODDYNMOOOYNMODDYNMODDYNMODDYNWOODYMMODDYNMODDYNMOODYNMODDYNMOODYMMOODYMMODDYNMODDYNMPDDYMMO'.lOYNMODDYNMODDYNMOOOYNMODDYNMOOOYN"ODDYNMODDYNMOODYNMODOYNMODDYNMODDYNMOODYNMODDYNHOD
15.01,
4014ft?40340440540640740R40941041141241341441S4164174184194?04?14?24?34?44?54?64P>74?84?94304314324334344354364 374384 3944044144?443444445446447448449450451
PAGE
10
15
35
c
76/74 OPT = 1 OOIJNO=*-»/ FTN 4,ft*433
SUP-ROUT INF Pt'JNFrMJD.L.N.P.ROTPlN.MIPPIN.KCROT.KFROT,. SOCFOT.sDFnoT)
cccccccccccccccccccccccccccccccccccccccccccccccccctcccccccccccccccccccccccccccccccccccccccc
CALCULATE THE COOPOINATF REDUCTION MATRIX PFOR * PINNED RIINnLE STRING,
INTRODUCE STIFFNESS AND STRUCTURAL DAMPING TFPMSASSOCIATED WITH THE CONSTPAINEn END.
INPUT...
RPIN a « ...FNO RUMPLE FREE (DEFAULT SFTTING)= 1 ...FNfl RMNC T PINNFD
CCCCCcccccc
JPIN = 0 ...INTERMEDIATE BUNDLES FRFF (DEFiULT SETTING)C= J ...END OF RIJNOLE J PINNFO
KCROT = ...FNP CONSTRAINT AOPITlnN TO VCST MATRIXKFBOT = ...FW CONSTRAINT APPTTION TO VFNO MATRIX
DC^OTX = ...FXTRA C«;T DAMPING FACTOR FOP FNO CONSTRAINTDF.POTX = ...FXTRA CONSTOAtNED FND PLATE DAMPING FACTOR,
( KCROT = KFBOT = DCBOTX s OEROTX = 0. BY OF.FAIILT.
ccccccccc
ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
ccc
ccc
ccc
ccc
LAMPS FUNCTIONS.
INTEGER ASSIGNC.COPY.01V.SCALAR.SMULT3.SUBMAT,SUBROW,TRANS.ZEROCOMPLEX FLEMENT
LAMPS STORAGE ADEAS.
INTEGER COL.HIP,ROWCOMPLEX SPACECOMMOM /M?/0IP/M3/P0W/M«/C0L/M5/LINK/M(i/SPACE
MATRIX POINTERS AND VARIABLES.
INTEGER AM.A ',flM,BZ,BPIN,DCBOTX.DEBOTX.R.SDCBOT.SDEBOTLOGICAL BOTPIN.MIOPIN
SKT nF.FAULT VALUES.
BPIN»JPIN*7ERO(1.1)KCB0T*KEB0T=0roOTX*0EP.0TX»7FR0 (1.1)JP*0
INPUT DATA.
C»LL BEADfeltHRPlN, SPIN,. «HJPJN, JPIN,. «iHKCROT. KCBOT,. SHKFf.OT. KEBOT.. 6H0CBOTX, DCBOTX.. ftHOFBOTX, DFB07X)
71-09-18
PINNEDPINNERPINNEDFJNNEOPINNEDPINNEDPINNEDPINNEDPIMNFDP1NHEDPINNEDPINNEDPINNEDPINMFDPINNEDPINNEDPINNEDPINNEDPINNEOPINNEOPINNFDPINNEDPINNEDPINNEDPINNEDPINNEDPINNEDPIMNEDPINNEDPINNEDPINNEDPINNEOPINNEDPINNEDPIMNEDPINNEDPINNEDPINNEDPINNEDPINNEDPINNEDPINNEDPINNEDPINNEDPINNEDPINNEDPINNF.DPINNEOPINNEOPINNEDPINNEOPINNEOPINNED"INNEOPINNEDPINNEDPINNED
15.01.5ft
?3*•5
678910111213
1516171819?0?1?2?3?*?"5?6?7?*?9303132333*353637383940«1• 24344454647484950515?535455565758
f-ASE
SUBROUTINE PINNED
70
80
35
74/74 OPT=1 BOUND»»-»/
i;.» F.AI.(FLF>«FNT<nPTNf \
FTN «.6*433
rF(ROTPIN.O!».MrOPINl GO TO 3(SO TO 10
cC CONSTRUCT THF COpPOIMATF REDUCTION MATRIX P.C
3 JP = MIF CIOPTU)JP=RFAL(FLFMFNT(JPIN,1,1))AM=JOINVtir>FNT(JP-1 ) .TPANS(NEG(DIV(SUBMAT(L.1.1 <
. , Sunoow(L .JP) ) ) ) )IF(POTPIM.«Nn..N0T.MlOPIN) fiO TO 8A 7 = 7 F P O ( M - J D . J P - 1 )
pIF(MlnPTW.AMn..NOT.BOTPIN) GO TO 6R 7 = Z F « 0 ( J P . N - J P - 1 )
. )) )) )P = A<;<;i(SNC(J0lKlH(J0lMV(AM,«7) .J0INV(B7,BM)>)ÏF(WIOPTN.AMD.BOTPIM) GO TO 7GO TO Ifl
7 CONTINUECC iNTRPOUCF FXTRA «STRUCTURAL DAMPING FND TERMS,C
sncBOTrA«;cîIfiNr(JOINv(7ERO(N-l (N) « JOINM (7ERO ( 1 tN-1). (SCA-LARin.Ot 1.0) tDCROTXtKCBOT) ) ) )<;nFBOT=A<;$ir,NC(JOINv<ZEfiO<N-),N) ,J0INH(7FR0U »N-1>. (SCALAR(n.O.l.O)tDEHOTXtKEBOT))))CALL SHOWMEH
10 RFTUBNEND
.SMULT3
.SMULT3
••09—1 fl
PI N'NFOPINNEDPIMNEDPINNFDPINNEDPINNEDPINVEOPINNEDPINNEDPINNFOPINNEDPINNEDPINNFOPINNEDPINNEDPINNFDPINNEDPINNEDPINNEDPINNEDPINNEDPINNEDPINNEDPINNFDPINNEDPINNEOPINNEDPINNEDPINNEDPINNEDPINNEDPINNEDPINNEDPINNED
16.nl.5R
6061636364656667686970717273747576777879RO818?838485R6RTASft9909192
PAGF
I
74/74 ORT=1 7R-09-1R 15.01.SB PARE
in
30
35
•SO
ss
A SUBROUTINE DESIGNFC TO CALCl'LATF THF TPANSTF'lT ANC STFADY STATE CPFSPONSE OF » L INF.AB STRUCTURE TO APPLIFH HAPPONIf FORCFS.
WHICH =
SUBROUTINE FORCFS(POTPIN.PPINV«CMAT«MIOPIN.HOOFS.wu.N.R,TC"AT> FORCESCcccccccccccccccccccccccccccccccccccccccccccccrccccccrcccrcccccccccccrccccccC INPUT.cccccC TMAXC SAMPLFC OMEGAC FORCEccC OUTPUT.cC RESPONSFcccccC STEADYC HAPMONICC PESPONSECc
TO HARMONIC FORCFS NOT RFOUIPFO,1 ...CALCULATE TRANSIENT HFSPOMSF.? ...CALCULATE STEADY HARMONIC RFSPONSF.1 /..CALCULATE TRANSIENT AND STFAilY RFÇPONSE.. . . . .TRANSIENT RESPONSE ORSFRVAT1ON TIME (SECONDS). C. . . . . T I M E INTERVAL BETWEfcN CALCULATIONS (SFCONOS). C. . / . .FORCING FPFOUENCY <H7>. C.. . . .COLUMN VECTOR OF FORCES APPLIED To PurvDLF FNPS.C
OF FORCE CONTAIN MAGNITUDE AMO PHASE INFORMATION,)CCCCCCccccccccc
.....COLUMN VECTOP OF COMPLFX N||MRFRS CAI.CHIATFOAT THF FNti OF FACH SAMPLF TIMF INTFRVAI . THF TIMFDOMAIN OFFLECTIOMS APE THF RFSPOK'SF TO ROTATINGVECTOP FOPCES. THF REAL AI»P IMAGINARY PARTS OFRESPONSE GIVE THE DEFLECTIONS COPRESPONDINP TO THECOSINE AND SINF COMPONENTS OF FOPCF RFSPFCTI VFI.Y.
..'...* COLUMN VFCTOR. DESCRIBING THF AMPL1TUDF AMPPHASE OF THE HAPMONIC BUNDLF FND OFFLFCTIONS.
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCcC LAMP* FUNCTIONS.c
IHTEGEP ASSIGN.ASS IGNC.CREATE.01DELETE.OUTPUTC.OUTPUTLtPAGE,. SCALAR,SMULT.SMULT3.SUB,SUPMAT,TOANS,COMPLFX
LAMPS APFAS.
INTFGFR COLtDIR.POWCOMPLEX SPACECOMMON /M?/nTP/M3/ROW/M4/C0L/M5/LINK/Mft/SPACE
MATRIX POINTERS AMD VARIABLES.
INTFGFR RRINV.CMAT.DISPLA,F,FRIG,FORCE,FBfCK.GENFOR,OMEGA,PHI,. POLA»,P,SAHPLE,TCMAT.TMAX»*IHICH,7,7EOCOMPLFX COMFflA.ELINT(60)»ElMULOGICAL «OTP|N,MIOPIN
SET DEFAULT VALIJFS.
WHICH»7EPO{1,Ï)
FORTFSFORCFSFORCESFORCESFORCESFORCESFORCESFOPCESFORCESFORCESFORCESFOPCF.SFORCESFORCESFORCESFORCESFOP.CFSFORCESFORCFSFOPCEsFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFOPCESFOPCFSFORCFSFOPCESFOPCFSFOPCESFORCESFOPCESFORCESFORCESFORCESFORCESFORCESFORCESFOPCESFORCESFORCES
4567
9inil12131415lh17
?4
?T?8?930313233343536373839404142434445464740
4950SI52535455565758
SUBROUTINE FORCFS 74/7* FTN 4.6*433
71)
100
no
cc
tMPMT DATA.
CALL PFAOM«;nuHTCH.. 4HTMA».
«.HqAMRLF.^HOMFRA.
WHICH,TMAX,SAMPLE,DMFGA,
LOf-IC=PFA|.IF (LOGIC.EO.O) GO TO 10jOi)Mwy=0liTPMTrC">wTMAx SAMPLE
. PAGE (jnTMH(TWA*.jnjMH(<:AMPLF.OMEGA] ) ) )( H 7 ) > 3 ,
C CALCHLATF TMr ' « T D K OF GFNEPAl I?FO APPLIED FOPCF« FOP 7HFC UMCniiPLFn F1ll«TInNS OF MOTION.C
FstSSIfiN (MULT (TCMAT .FriPCE) ).DFLFTF(F) ) )
pGFNFOS**S'iIfiNC(MliLT3( INVF.PT (MODES) .BRINV.FBIG) )IF tLOr . IC .EO.?) 00 TO 9
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCc cC CALCULATION OF TPAiYSIFNT PESPOMSF. CC THF PF.<?IILT«5 APF TIME DOMAIN OFFLECTIONS. "CC Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
Tno=pEAL (FLE'^FNT(SAMPLE, 1 , i ) )
CniiTtMitFPO 7 1 = 1 . U KFLMi]=F.LF»'FNT(Mti,I.l)
.((0.0,1.0)»(COMFGA-ELMU))7 CONTINUE
TNTIMEin(CREATF(EI.INT,?»N,l,?»N) )
CALCULATE THE AMGIIL»R VELOCITY AND ANGULARMATR1CFS AT TIME TOP.
(0 .0 . 1 ,0)»ELM|I»TOP) ) /
7=ASSir,nr(MHLT3(MOnFÇ,INTIME,GFNFOR) )PHl=AS<5IGNC(J01NH(DeLFTE(PHI),SUBMAT(7.N»ltl,Nfl)))
IF(TOP,RF.REALfFLFMEMT(TMAX,l,l))) GO TO ?C,n TO I
? CONTINUFP.MlDPIN) PHI*MULT(P,OFLFTe(PHI)
ÏMULT (CM»T,PHH
C OUTPUT THE TIME DOMAIN RESPONSE.C
jOIIMMYxOUTPMTL(?7HTPANSIENT HARMONIC RESPONSE.3,DISPLA)CALL SHOWMEM
7fl-0<)-I8
FOPCFSFOPCESFORCESFORCESFORCESFORCESFOPCESFORCESFORCESFORCESFOPCFSFORCESFORCESFORCESFORCESFOPCESFORCESFOPCESFOPCESFORCESFORCESFORCESFORCESFORCESFORCESFOPCESFORCESFORCESFORCFSFOPCESFORCESFORCESFORCESFOPCESFORCESFORCESFORCESFORCESFOPCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFOPCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCES
15.01.SA
59f>0AlA363A4A56667f.86970717?717*757S777879BOCl8?S3a*85S6R7flflR990919293949596979899
- 100101102Î03104105106107108109110111112113114115
PAGE
SHPROUTINE FTN 4.6.433 1 5 . n i . S A PAGE
130
135
1*0
CALL PFI_FTFX|TMTTuF.r>)CALL CLFANUP
cccccccrcccccccccccccccccccccrccccccccccccccccccccrccrcccrccrccccccccccc STFAOV STATF HAPMONTC RFSPONSF.
THF »E<;Ul T IS A MATPIX OF VIBRATION AMPLITUDE ANH OHASFSCOPPFÇPONOTNG TO THF LATFRAL niSPLAfFMfNTS OF THF PUNOLF FMOS.THF FRfnuFNCV OF THF PFSPOMSE IS THAT OF THF APPLIFn FORCFS.
c cccccccccccccccrccccccccccccccccccccccccccccccccccccccccccccrccccccccccc
IF(LOfiir.FO.l) GO TO A<> CONTINIIF
C L S I. n(SMIILT(SCALAR ( 0 . 0 , 1.0) ,»»ll> ) ) )rFnïASSIRNL^MiiLTS (MOOES, INVFPT»FBFCK(,GFNFOP))PHI=ASSIf iM(SUHMAT(7F0,N*l , l .N, l )>IF(HOTPTN.OP.MIOPIN)PHI=ASSIGNC(MULT(P.OFLFTF(PHI)))POLAPSA<:SII;NO(MIILT(CMATPHI) >
CC OliTPHT THE STEMOY HARMONIC RESPONSE.C
jni.'MMY=O(lTPUTLC?*HSTFADY HARMONIC RFSPONSF.3.PAGF (POLAR) )8 CALL OELFTFMF.O)
CALL CLFANIIP10 CONTINUE
CALL SHOWHFMRETURNEND
FORCESFOPCES
FORCESFORCFSFORCFSFOPCTSFORCESFOPCFSFOPCESFODCFSFflPCFçFOPCESFORCESFORCESFORCESFORCESFORCESFOPCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCESFORCES
11».117
1??1?31?*
13013113?13313*13513*137138131»1*01*11*21*3!**1*5
74/74 OPT=1 ROUND»»-»/ FTN 4.6*411 7H-09-18 15.01.58 PARE
10
?0
•5
•SO
A SUB«»OIITINF DFSir.UFD TO CALCULATE THF RFSPONSF OF fl STRUCTURE TOAPPLIFO PAN00» FORCES.
INPUT.
TWO SFTS OF (-AROS APF REOIIIPEO FOP THIS SURROUTINF.
SUBROUTINE RJ,k'D0fMWnTPIN,nRINV.CMAT.MI0PIN,MODFS.'<l'.N.P,TCMAT)CccccccccccccccccccccccccccccccccccccccccrccccceccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccrcccccccccccccccccccccccccccccccccccccccccccccccccccC L»MPÇ FUNCTIONS.c
INTEGER At>O.ASpIGN.ASSI6NC.C0NJ,O,nELC0l,DELETE.DIAG.GFTCOL,. OUTPUT,OUTPUTL»°AGE,POWER,SCALAR,5MULT,SMULT3,SUB,SURCOL.
SURMA»,SURPOW,TRANS,ZFROCOMPLFX FLF.MENT
THF FIRST OFSC^IOFS THF NATljPf OF THE APPLIFO FORCFS.nATA = 0 RFSPOMSF TO RANDOM FORCES MOT RFQUIRFD.
= 1 PFSPONSF TO niSCPFTF FnPCF «;PFCT»A* ? RESPONSE TO AVERAGED POWFP SPECTRA= 3 PFSPOWSE TO UNCORRELATFD fORCF SPECTRA RFOUIPFD.
POINTS = .......NUMBER OF SPFrTPAL L1NFS IN THF FORCF DATA.FMIN = .. IOWFST FPFOIIFNCY IN THF FOPCF SPECTRUM.
RANWin = .......FPFOI'FNCY INTFPVAL RETWF.FN SPFCTPAL LtNFS.
THE SF.CONO CAR(1 <;FT CONTAINS THF APPLIED FORCF DATA.1. FOR DATA a 1, WF INPUT OFORCF (NEWTONS) A SINGLF NXM MATRIX
OF M CPMPLF» FOPCF VAIUFS, ONF FOR FACHFNO, ^0R EACH OF M SPECTRAL LINES.
?. FOR DATA = ?. WF INPUT PSPFOP (NFWTONs«#?/H7) A NxN. OF AUTO AND CROSS POWER SPFCTRAL OFNSITIFS OFAPPLIFP FORCE FOP EACH OF M SPFCTRAL LINES.
•3. FOR OATA = 3. WF INPUT UNCORF (NEWT0NS»»?/H7) A SINGLE NXMMATRIX OF AUTO POWER SPECTRAL DFNSIT1FS OFAPPLIED UNCORRFLATED FOPCFS.
OUTPUT.
THF INFORMATION PPOVIDEO AS OUTPUT CONSISTS OF...1. THE AUTO DHWFP SDFCTPAL tlFNSITV OF RESPONSE (MFTRFS«»?/H7)?. THF. MFAN SOUARE VALUE (MFTRES»»?) AND ROOT MFAN SOUARF VALUF
OF THF PESPONSE AT EACH RUNOLE FNO.IN THE CASF OF TtF DISCPETE FORCE SPECTRUM INPUT(OATA=1)«THE
OUTPUT IS...1. THE COMPLEX RESPONSE SPFCTRUM(METRES),?. THE AUTO POMFR SPECTPUM(METPES»»?),3. THE MFAN SOUAPE VALUE AND ROOT MEAN ÇOUAPE VALUE OF RESPONSE
FOR EACH RUNOLE END.
LAMPS STORAGE APPAS
INTEGER COL,DIP,ROW
ccccccct
r.CCCC
cccccccccccccccccccccccccccccc:c
RANDOMRANDOMRANOOMRANDOMRANDOMRANOOMPANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANnOMPANDOMRANOOMPAMDOMRANDOMRANDOMRANDOMRANDOMRANDOMPANDOMRANOOMRANDOMRANDOMRANDOMPANDOMRANDOMRANOOMPANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRINOOMRANDOMRANDOMRANDOMRANDOMPANDOMRANDOMRANDOMRANOOMRANDOMRANOOMRANDOMRANDOMRANOOMRANDOMRANDOM
?3«56789101112131*1516171119?0?l?2?-\?4?S?6?7?8?930313?333435?637TB3940414?4344454647484950SISH535455565?58
SUBROUTINE RANDOM 74/74 O°T=l ROUNDS*-»/ FTN 4.6*433 7R_O"5-lfl 15.01.SB PA<" .
70
lno
110
COMMON /M?/ •»D/M1/RnU/M4/rOL/Mc5/LINK/fft/çP«ri r
cC MATRIX pnlNTFCx; AND " A R I A P L E S .
cINTF.GFR AIITOPY.RANWJp.FIRJNV.CHAN^F.CMAT.PATA.OFORCF.niSMAT.DISPOW,
. DISVFr.OMSTAR.DPOWFR.FM IN,FORGFN.POINTS,PSDFOP.R.RMS«ES.TCM AT, TRAMAT, TSTAR.TSTART.IINCORF
LOGICAL nOTPTN.MIOPINCC SET nFFAULT VAL'IFS.C
DATA=7FRO(1.1)CC InIPUT OATA.C
CAUL PFAD4(4Hr>ATA, DATA.POINT?,
. 6MPANWIO, RANWIO)CC CALfl'LATF PFSPONïF M A T P I X FOR EACH SPECTRALC
LOROAT=RFAL (F|.FUFNT (DATA, 1 , 1 ) )IF iLOGOAT.F.O.n) Gd TO 310M=PPAI. (FLFMFMJ (POINTS. 1 , 1 ) )NM=GFTCflL(CMAT)ISPEC=1
çC FVALUATF THF TRAN^fER FONCTION MATRIX FOR FACH t;PFCTRAL LINE.C
MATFOP=J0IMV(7FR0(NN,NN),TCMAT)IF(R0TPI'J.OR.MtnPIN)MATFOP=JOINV(7f PO(M.NN) ,MML T ( TRAN*? (P) .TCMAT) )FOpfiE^=A<;si'îN(: (fiiLT3(rwveoT(MonF<;i . P P ^ V , M A T F O R I I
1 CONTINUACHANGE*A'ÎSl'îNCCîllWlSMULTK^CAL AR ( 0 . 0 .6 . ?P3?> .FMIN, lOFNT (2»N) ) ,
. [USMI/^TI SCALAR (!/. 0 , 1 . 0 ) «Ml)) ) ) )MATPHI = ASSlGNC(StlBMAT(MuLT3(MOOES, INVERT (CHANGE) .FORGFN) .
N»1.1,M,NN))IF(ROTPIH.no,MInPIN)MATPHI=ASSIGNC(MULT(R»DFLFTE(MATPHI)))TPAMAT=ASSIGNr(«ULT(CMAT.MATPHJ))
CC LOGIC CONTROL F0O DATA.C
IF(LOGOAT.GT,?) GO TO 210IF(LOGDAT.GT,l) 00 TO 110
Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc cC CASF. 1 . CC CALCULATION OF RF.SPONSF TO DISCPFTE FORCE SPECTRA. CC Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccceccccccccccc
IF(IJK.FO.l) GO TO 1?3IJK«1
RANDOMPANDOMKANDOMPAK'DOMPANDOMRANDOM
PANDOMPANDOMRANDOMRANDOMRANDOMRANDOMPANDOMRANDOMRANOOMRANDOMRANDOMPANDO»RANDOMRANDOMRANDOMPANDOMRANDOMRANDOMRANDOMRAMDOMRANDOMRANDOMRANDOMPANDOMRANOOMRANDOMRANDOMRANOOMRANDOMRANDOMRANOOMRANOOMRANDOMRANOOMRANOOMRANOOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANOOMRANDOMRANOOMRANDOMRANOOMRANDOMRANDOMRANDOM
F»96061f,2A364
Aft67686970717273747576777879BOSIB283B485B6B7R8H990919293949596979399
100
ml1021 "310410510610710810911011111?113114115
ooI
SIIPROHTINF RANOO" 74/74 T=l ROIJNO=*-«/ FTN 4 .6*433
130
135
1*0
i*?
150
160
165
170
CALL RFAni(6NOFOPCE.0F0RCF)1?1 CONTINUE
OI'^VFCsA^'ÎIRNCCULT (TRAMAT,SIIPCOL (OFORCE. ISPFC) ) )
cC LOOP PACK TO I Tp CALCULATE THE DISPLACEMENT VFCTORFOP THF NFXT SPECTRAL LINE.
FMIN=ASSIPNC(AOD(OELFTE(FMIN).PANUID))nis»AT=A?:FIBNc'(JOlNH(0ELETF(niSMAT) .OISVFC))
IF(ISPFC.GT.M) GO TO 1?BO TO 1
12 CONTINIIFDISMAT=ASSI(5NC(OELCOL (DELETE (DISMAT) ,1))
CC THF COMPLEX DIS°LACFMENT RESPONSE MATRIX DISMAT HAS NOW
BEEN EVALIIATEn. OUTPUT THE RESPONSE SPECTRA.C
POINT 1113 FORMAT(lHl)
DO 14 1=1.NNPRIMT 15,1
15 FORMATncH PFSPON«;F SPECTRUM AT END OF PUNOLF ,12)jnilMMyzOIITPllTtSURROwiDISMAT,!) )
1* CONTINUE
cC CALCULATE THF AUTO POWER SPECTRUM OF RESPONSE.C
nM«STAR»CONJ(OISMAT)OPOWEP = A<;<;iGNC(SMllLT3(SCALAP(0.5,0.0) .DISMAT.DMSTAR) )
CC OUTPUT THF AUTO POWEP SPFCTRUM OF RESPONSE.C
PRINT \f>16 FORMAT(IHl)
00 17 J=1.NNPRINT 1«.J
18 FORMAT (SCH "F.AN POWFR SPECTRUM OF RESPONSE AT F.NO OF BUNDLE .12)jnuMMY=nilTPIJT(SUP<ROW(DPOWFR»J) )
IT CONTINUF
cC CALCULATE THE WÇAN «ÏOI.IARE DISPLACEMENTS AT THE BUNDLE ENDS.C
19
C EVALUATE THF P.M'.S.VALUES OF RESPONSE.C
RM«!RES*POWÇR(MsniSP, <O.5tO.O>)JDUMMV«OUTPIiTL«*0HrO0T MKAN SQUARE RESPONSE AT BUNDLE FNPS.5.. PA(5E«»M«:PFS>)CALL OELFTFX(FOPGEN,OICALL CLEANUP
l00 19 K=1,MMSDISP»ASSir,MC(AOO(OELFTE(MSDISP) .SUPCOL (DPOVER.X) ))CONTINUEjnilMMViOUTPUTL(35HMF.AN SQUARE RESPONSE AT BUNDLE FNDS.4,
P A G E r p
-no-18
RANDOMRANDOMCAMDOMRANDOMRANDOMRANDOMPANHOMRANOOMRANDOMRANDOMRANDOMRANDOMRANOOMRANOOMRANDOMRSNPOMRANDOMRANOOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMPANOOMRANDOMRAMDOMRANOOMRANOOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANOOMRANDOMRANDOMRANDOMRANOOMRANDOMRANDOMRANDOMRANDOMRANDOMRANOOMRANOOMRANDOMRANDOMRANOOMRANOOM
15.01.5R
116117111119l?01?11?21?31?*1?51P61?71?81?913013113213313*13513613713B1391*01*11*21*31*41*514614714814915015115215315*15515615715815916016116216316*165166167168169170171172
PAGE
I
O
SUBROUTINE RANDOM 74/74 OPT=1 ROHND=*-»/ FTN 4,6*433 7B-0Q-lfl 15.01.5fl PAGE
1B0
1*5
100
195
?00
00 TO 3in110 OONTINIIF
Ccccccccccccccrcrcccrcccccccccccccccc<"cccccccccccccccccccccccccccccccccc cC CA«E ?. Cr CALCULATION OF RFSPONSF TO AVERAGED P.S.O. OF FORCE. CC Cccccccccccccccrcccccrcccccccccccccccccccccccccccccccccccccccccccccccccc
CALL REAni(ftHPSDFOB.PSDFO»)cC CALCULATE THF AUTO POWFO SPECTRA OF DI^PLACFMFNTS.c
TSTART=TPAN9(rONJ(TRAMAT))AIITOPY=AÇSIGNC(OTAfl(MULT3(TPAMAT.PSDF0R,TSTART> ) )FMIN=ASSIGWC(ADD(DEL(;TF (FMIN) .BANWIO) )niSPOUsASSIGNCUOINHIDELETECDISPOW) .AUTOPV) )ISPEC=ISPFC«1IF(ISPFC.GT.M) GO TO 3?GO TO 1
210 CONTINUFCccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc cC C»SF 3. CC CALCULATION OF RESPONSE TO UNCORRELATFD PANHOH FORCFS, Cc cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
JFIIJKL.FO.l) GO TO 1234IJKL=1CALL REAni (iSMIINCORF.UNCORF)
1234 CONTINUECC CALCULATE AMTO POWFR SPFCTRAL DENSITY OF RFSPONSE.C
TSTARsCONJI TRAMAT)AUTOPY = ASSIfi*IC'("ULT (5HULT (TRAMAT,TSTAR) «SURCOL (UNCORF.ISPEC) ) )FMIN=ASSTGNC( APP (DELETE (FMIN) ,BANWID )OISP0w=ASS!i-\IC ( JOINH(DELETE (OISPOW) .AUTOPV) )1SPEC=ISPEC»1IF(ISPFC.GT.M) GO TO 3?GO TO 1
32 CONTINUEniSPOW*ASSI<5UC (OELCOL (DELETE (OISPOW) . 1 ) )
CC OUTPUT THE AUTO POWEP SPECTPUM OF RESPONSE.C
PRINT 3333 FOPMAT(lHi)
DO 34 ITz l .NNPRINT 35.11
35 FORMAT (54H AUTO POWER SPECTRUM OF DISPLACEMENT AT FNO OF RIINDLE •I?)
JOUMMY*OIITPIJT (SUBBOwtDlSPOVI.il.M34 CONTINUE
RANDOM'ÎANOOMPANOOM
RANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRAN:')OMPANOOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOMRANDOM
17317417517617717*179lfiO
lf>3
1P919019119a193194195
198199?no?ni202?03?04
?09
214
I
o
2?5
2?9
SUBROUTINE BANPOW 7* /74 OPT=1 ROIJND = » - » /
74/74 FTN 7H-0«-ln 15.(11.59 PAGE
SI IP «OU TT MF 3Fr"0R((:inTPIkJ.POINv,C"«T.JP.MinPTMtMOnF^.«li,N,R,TC»AT)ccccccccccccccccccccccccrcccrcccccccccccccccccccccccccccccccccccccccccc
10
?0
30
•SO
crcccccccccccccccccccccc
A SUR91IJT7NF DFÇIfiNFn TO CâlCULiTE THF TP/SNSFFPOFCEPT4NCF ANP "OBIUITY BETWEEN POINTS ON T*F STRIICTIIBF OVFP
A SPFCTFTEn RANfiF OF FPFOFMCIES.
TWPMT.
SHAKFR = . .'. ..NllMHFB OF THE RUNflLF IT THF FND OF WHTCH THFFOBCF IS OPPLIEO.
(IF SHAKFP = 0, TRANSFER FUNCTIONS APF NOT CALCUL'TEn.)FMIN = .....LOWFP FOFOIIENCY LIMIT OF THANSFFP. FlIK'CT IONS.FMAX = .....UPPFP FRFOUENCY LIMIT PF TRANSFFR. FUNCTIONS.
BANWTO a .....FOF.01IENCY INTERVAL RETWEEN CALCULATIONS.(ALL FPEOUEMCIES ARF IN H7.)
OUTPUT.
RECFP = .....COLUMN VFCTOPS OF PFCFPTANCE (0/F).MOfilL = .....COLUMN' VFCTOPS OF MOPILITY (V/F).
(EACH VFCTOP DFSCPIPES THE AMPLITUOE AND PHASE OF THETRANSFFR FUNCTIONS AT THF PtlNOLE FNDS WITH PFÇPFCT TOTHF PSFUO0 FORCE INPUT.)
CCccccccccccr
CCCcccccccc
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC'CCCCCCCCCCCCCCCCCCCCCCCCCCcC LAMPS FUNCTIONS.c
INTFGER ADD.ASSIGN.ASSIGNC.CREATEtO.DELETE.GF.TROWtOIITPUTCtOUTPUTTtRAGF.SC 4L AP,SHULT,SMULT3. SUB. SUBHATt TRANS. 7.EP0
COMPLEX ELEUFNT
LAMPS STORAGE ASFAS.
INTFGFR COL.OTR.ROWCOMPLEX SPACFCOMMON /M?/DIP/"3/ROW/M*/COL/H5/LINK/Mft/SPACF
MATRIX POINTFRS ANP VARIAPLES.
INTFRFP flANWlP.PRINV.CHANfiE.CMAT.FMAX.FMIN.PHASFM.PHASFR.PHIMOB,. PHIPf:c.P.RECEP»SHAKF.R,TCMAT.UNFBIG.UN6FNF,IINIFOR,7F.PNEWCOMPLEX CP.«<»(30).PR(30) .CM,MM (30) ,PM(30)LOGICAL
SFT DEFAULT VALUF'-:,
SHAKER=7EBO(l,l)
1WPUT DATA.
CALL REAn*(fiMSHAKF.R, SHAKEP,«HFMlN, FMIN,
. 4HFMAX. FMAX,
REC"ORRECMORRECMORPECMORRECMORRECMORPECMORREC"ORRECMORRECMORREC'ORREC'ORRECMOBRECMORRECMORRECMOR
RECMORRECMORPFCMORRECMORRECMOBRFCMOBRECMORRECMORPECMOBRECMOBRECMOBRECMOBRECMORPECMOBP.ECMOBRECMOBRECMOBRECMOBRECMOBRECMOBRECMOBRECMOBRECMOBRECMOBRECMOBPECMOBRECMOBRECMOBRECM08RECMOBRECMOBRECMOBRECMOB
è34567891011121314ISÎ617IB19?0
?3
?7?8?9303132333*35363738394041424344454647AR49SO5152S354S5
58
RFCMnc 74/74 FTN
90
100
110
OF r.FNFOBCF OF (INJT MAGMJTIJPF.
! n K , c D . i i | no m hT F ( K , F O . . | P ) fin Tn 1
CALL S T O " F ( i i » i j F n o . K , i , ( i . 0 i n . n ) )
7( IMVEPT (HDDES) .(
. PAPF (JflTMW i'.Jnl^'4(<|JAKFn.FMINj , JflIMH2 C1NTINIIF
RANWID.*,) ) )
. 0 . 1 . 0 ) , M ( J ) ) ) )
INVEST (CHANGF)
Î f R.PFLiTTF (PHIMOP) ) )) )
IF :';Mnt>IRFCFP = MiiLT (CM
: CALCULAIc
•"•i.ô rHASEf, OF THE TRir.sFEii FlINCTlUfis.
DO 7 J= lCPïtLFWFMR(J)=C"°
, J . l ). 0 . 0 )
MlM M ( J ) = C " P L X ( C A O < ; ( C M ) , o . O )
7 CONTJMIIF
PHASFR = ASSir.N(CRFATE (PR.NDMBEP.l .NUMBER) )
PHASEM=ASSir,Me'(CREATE (PM.NMMPER.l,NUMBER) )CC OUTPUT THF. TOANSFFP FUNCTIONS.C
POINT Çç FORMAT(///l'5X,inHPFCePTANCF.54X.8HM0BILTTV)
jnuMMY = ntlTPIITCnt;HMAGNITUDE PHASE MAGNITUDE PHASE.*.. J0INHI jnjNH(HODRFC.PHASEI') . JOINH (MOOMOP.PHASEM) ) )TALL PFLFTF.X(ewANfie,O)CALL CLFANUPFMINsAS^TGMCIAnOlOELETEIFMÎNJ.BANWID))rF(»EAL(FLFMENT(FMlN. l . l ) ) .6T.Bfc-AL(ELEMFNT(FMAX,l . l ) ) ) fiO TO 3GO TO 2
î MNTINUFDRINT 4
7fi-n a
p F r K , n R
Vff»nu
Bi-CMilH
HETMORRF.CMORRECORKFCMflRHEC»ORRf-çuOPRErcnRRECMORRFCMOHRFOORRECMOHRFCOqBEC1HRECMORR( CMORRFCMORRECMOMPECMORRFT.MOR
RFCMOBRECMOljWECOBRECMORRFC"OR(<F.CiÛi'iRFCMORHECMOURECMOR(VEC^OBPFCMQR
il) RECMORRECMORRECMOB
0) RL'CMO'îRECMORPFCMORRECORRECMOBRECMOBPECMORRECMOBRECMORRECMOBRECMORRECM08RECMORRECMOBRECOflHECMORRFCMORRECMOBRECMOHRECMOBRFCMOR
15.f. 1.5R
CQ
f-nf-lh?ft3f4f.5tf,
f.7Aflf,970717?73747176777879POni
P304«5floB7n4P990019293949 b96979B99100101102103104105106107108109110111112113114115
PAPE
<;il«»OIITINE pFCMfiP 74/7*
1 1 ^ 4 F n D i i A T < / / / i n x . 31 Cnt-'T
rail
r= l R(lUN0=4-o/
FS AND
FTN 4.*»<
4RF 7FRn.i
CALL SHOWMFMRFTIJ"NFNR
7fi-oa-lfl 15.01.55
BE CoBRECOHRFCwnnRF<;MOR
r iECORRECMOBRECM08RECMOS
13 fei nl i a119l?o1?11?21?3
PA6f
I
o
I
74/74 OPT=J ROUND=*-»/ FTN 4.6,433 15.01.58 PAGE
SUBROUTINE CHLPOPIROTPIN.BOINV.CMAT.JP,MI .Mil.N.P.TCMAT)
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
CALFORCALFOR
10
40
cccccccccccccccccccccccccccccccccccccccccccccccc
A SUBROUTINE nFSJRNFD TO CALCULATE THF N'ATURF OF RFSOLVF.O FORCFSACTING ON A «5TPMCTURF.
INPUT.
THREE SFTS OF OATfl CAPOS APF PEQIIIPFD FOP THIS SUBROUTINE.
ccccccccccccccccccc
THE SFCO»iO HATS C4T> SFT IS OPTIONAL. FOP FXPEP=O,THF SECOND DATA CCAT) SFT IS MOT OPQUIRFD. FOR FXPFR=1.WF INRIJT MATRIX TRAMAT. C
TRAM4T = .. .T"ANSFF« FUNCTION MATRIX (»FTPFS/NFwTON).TRAÇAT IS A CNXM MATPTX f i n F4CM OF M SPKTTRAL I.INFS. C
cccccccc
THE FIRST OATH CAOO CONTAINS THF FOLLOWING FIVF V,TYPE = 0 .....FORCE CALCULATIONS APE NOT RFOMIDFD,
= 1 OP ? OTSCRETF FORCE SPECTRA RFOIIIRFp.= 3 OR 4 AIJTO AND CPOSS POWER OF FORCES REOUIPEO.
FXPFR = n .. ...TPAMSFFR FUNCTION MATRIX IS CALCOLATFn.= 1 FXP^PIMENTAL TRANSFER FUNCTION MATRIX Is USFO,
POINTS = ...NIIMOFP OF SPFCTRAL LINES JN THE DISPLACEMENT DATA.FMIN = ...LOWFST FREQUENCY IN THE FORCE SPFCTRUM.
BANWTO = . ..FBc1llF^CY INTERVAL BETWEEN SPECTRAL LINES.
THF THIPO naTA CAPH SFT CONTAINS THF 01SPLACEMENT DATA.FOR TYPE = 1 OR ?.WE INPUT 01SMAT(MFTPES).
niS»AT IS 4 NXM COMPLEX DISPLACEMENT MATRIX OF NOlSPLSCfFMT VALUES (ONE FOR EACH RUNDLF END) FOP FACH OFM SPECTRAL LINFS.
(FOH TYPF=?, CPOSS DOWFR SPFCTPA OF FORCF ARE CALCULATED.)FOR TYPE = t OR «.«F INPUT 01SPOw(METPES»»?/H7).
niSPOW TS A NXN CROSS POWER SPECTRAL DENSITY MATRIX FOR E»CH COF M SPECTRAL LIMES.
(FOP TYPE = 4. CROSS POWER SPFCTRAL DENSITIES OF FORCFARE CALCULATED.)
OUTPUT.
THE OUTPUT CONSISTS OF A MATRIX OF PFSOLVFD FOOCES.(IF. IF THESF FORCFS APE APPLIED TO THE IP PESPECTIVE BENDS,THF PFSPONSF wILL PF THAT SUPPLIFD AS INPUT DATA.)
FOR TYPE : 1 0» ?,THE FORCE ÔATA APF COMPLEX DISCRETE SPECTRA. CFOR TYPF S 3 OR 4.THE DATA ARE P.S.D.OF FOPCF, C
CCCc
IN ALL CASFS THE MFAN SQUARE AND ROOT MFAN SQUARE VALUESOF FORCES APF PROVIDED.
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccC LAMPS FUNCTIONS.c
INTEfiER ADD.AS.SK,N,ASSIGNC.CONJ,0,DELCOL.DELFTF.DFLPC,DELPOW,DIAfi,OETCOL,ONE,OUTPUT,OUTPUTL.PAGE,POwEP.SCALAR.SMULT,
CALFORCALFORCALFORCALFORCALFORCALFORCALFOR
CALFORCALFOPCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFOPCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFOPCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFOPCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFOPCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFOfi
345678910
n1213141516171819?0
?3?4
?7?B?93031323334.153637
4042424344454fc474849505152535455565758
SUBROUTINE CALFO» 74/74 OPT=1 ROUNDst-»/ FTN 4,6*433
65
70
100
105
110
SMULT3. SU".SUBCOL.SIIPMAT.SUPPOW, TRANS,7EB0
LAMPS STORAGE AOF«c
INTFfiFR fOL.DIP.POwCOMPLFX «PACECOMMON/M?/tHR/MVP0u/M4/C0L/M5/LINK/Mf,/sPACF
MATRIX POINTFBS AMD VARIABLES.
INTEGFP AiiTOPF.BANWIOtPBlNV.CHANGE.C»AT.nFORCF.OIS»'AT.riTSPOW,. EXPFR,FMiN,F0NE,FpBGEN,FORPOW,FOPVFC,FpOWFP,F<;TAO,. POINTS.PSOFOP.BtRMSFOP.TCHAT.TRAMAT.TSÎART.TYPFLOGICAL ROTP1N.MIOPIN
SET DEFAULT VAUIFS.
TYPE*ZERO(1.1)
INPUT FIRST DATA CAPO.
CALL PFAD5(4HTYPF.. TYPE,. 5HFXPER, EXPER.. 6HP0INTS. POINT"?,, 4HFMIN. FMÏN,, 6HRA.NVIf>, fUNWIO)LORTYP=PEAL tEL FWENT(TYPE,1,1))IF(LOGTYP.FQ.n)fiO TO 10LOREXP=REAL(FLFMENT(EXPER,1,1))MaREAL(FLFMcNT(POINTS»l,H>NN=GETCOL(C"AT)ISPEC=1nFORCE=FORPOW=A«;sIfiNtZERO(NN,H )
EVALUATE THF TPANSFEP FllfJCTION MATRIX FOR FACH SPECTRAL t INE.
MATFOR=JOIN'V (7FR0 (NN,NN) ,TCMAT)IF(HOTPIH.r>R,MjnPIN)MATFOP«JOIN«(2ERO(N,NN) .MULT (TPANS (R) .TCHAD )FPR(5FN=ASSTGNf (MiiLTlf INVERT (MOOFS) .PRINV.MATFOP) )
Î CONTINUEIF(LOGEXP.LT.l)fiO TO ?CALL BEA01(ftHTRSMAT,TRAMAT)
2 CONTINUE= ASSIGNCr^i(P(SMULT3 (SCALAR (0.0,6.?P32) .FMIN, inEMT (?»N) ) .O(SMIILV(SCALAR(0.0.].0),MU>) ) )
= ASSIf!NC(SUflMAT(MULTa (MOOES. INVERT! CHANGE ) .F0»f iEN) ,. N * l . l . N . M N ) )
IF(ROTPIN.OR.MinPIN)MATPHIsASSIGNC(MULT(R,OELETE(MATPHI)))TRAMATxASSIGNe(MULT(CMAT,MATPHI))TFIB.OTPINITRAMAT^ASSIGNClOELRCJOELETECTRAMATj.NN.NN))IF(MIOPtN)TRAMAT«ASSIGNCinFLSC(DELETE(TRAMAT).JP,JP))
LOGIC CONTROL FOP RESPONSE DATA INPUT.
IF(LOGTYP.GT.»)GO TO 6
7R-O<3-1R
CALFOPCALFORCALFORCALFORCALFORCALFOf:CALFOPCALFORCALFORCALFORCALFORCALFORCALFORCALFOPCALFOPCALFOPCALFORCALFORCALFORCALFOP.
• CALFOBCALFORCALFOP.CALFOP.CALFORCALFORCALFOi»CALFORCALFORCALFOHCALFOPCALFORCALFORCALFOBCALFORCALFOBCALFORCALFORCALFORCALFOBCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFOBCALFORCALFORCALFORCALFOR
15.01.SO
5960M*•?*3A46566«7f.B(,970717?73747576777R79BOPI
azft3A4ASA6P7B8A49091929394959697Q«
99
ino101102103104105106107108109110
in11211311*115
PAGE
o00I
«IIRPOHTINF CALFOR 74/74 OPT=1 POUND*»-»/ FTN 4.6.433 7B.0Q.ia 15.01.58 PAGF
11"
\?n
135
1*5
15n
155
160
165
1T0
CCCCCCCCCCCCCrXCCCOTCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCc cC CASF 1 CC rALCMLATIOM OF DISCRFTF FOOCF 5PFCTRA. Cc cCCCCCCCCCCCCCCCCCCCfCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCc
IF(IJ.F0.1)R0 T O 3TJ=1CAI.L
Pot'lT 3131 FnDMAT(lHl)
POINT 3P.ISPF.C3? FnPMAT(5?H CPns
IFrBOTPTM)P»lMT•î F0«"4T («^H OF.nil fn CPOSS POWER MATRIX.
If (MIf)PIN)PPTNT 36, JPFmrcFn cooss POWER MATRIX.
,1?)
CONTINUEFpPVFCïâ^'îIGMCIMlILT (INVERT (TRAMAT) .RUPCOL (DISMAT < I SPEC) ) )!F (LOGTYR.EQ.l ) GO TO 30
CALCULATE THF POWFR SPFCTPUf OF FOPCF. (NFWTONS»»?) .
.0.fll .FORVFC.lW (NtN) )
POWER SPFCTRUM OF FOPCF AT SPECTRAL LIWF NO. .12)
flOTTOM PINNF.P.)
PINNFO AT BOTTOM O
30CALL CLEANUPCONTINUE
LOOD RACK TO 1 TO CALCULATE THF OISCPETF FOPCE VECTOR FOo THE NFXTLINE.
FM I MsOSM'ïNCKnrK DELETE (FMIN),RANwID))IF (HnTPT'UEIWVFCsfl^IGNC ( JO INV (DELETE (FOPVFC) .7ERO11 . 1 ) ) )IF (« I OPT H) I0l|MMV = JOINV(SURMAT(FORvEC.l.l.JP-l . 1 ) »7EP0 ( 1 , 1 ) )IF(«IOPIW)FO«VEC = A<;siGNC (JOIMV(inuMHV.SUflMAT(OELETF(FOPVEC>
. J P . l . M N - J P . l ) ) )F = a?SIfiNC( JOINH(DELETF (OFOPCE) .F0PVF.C) )
IF(IÇPEC.GT.MjfiO TO 4GO TO 1CONTINUE
= AS<ÏI<5Nr fOELCOL (DELETE (OFORCE) .1 ) )
OUTPUT THE OISCRFTF. FOP.CE SPFCTRA.
POINT 41«I FO«MAT(1H1)
DO 42 1=1,NMPRINT 43.1
43 F0RMAT(4?H DISCRFTE FORCE SPECTRUM AT END OF BUNDLE il?)JOUMMYxOUTPUT |SURROw (DFOP.CE till
•? CONTINUE
CALFORCAtFOHCALFORCALFOOCALFORCALI-ORCALFORCALFORCALF03CALFORCALFOPCALFOrtCALFORCALFORCALFORCALFORCALFOHCALFORCALFORCALFORCALFORCALFOPCALFORCALFORCALFORCALEORCALFORCALFORCALFORCAI.FORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFOHCALFORCALFnnCALFORCALFORCALFORCALFORCALFOP.CALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCAlFOS
11^11711B119l?01?11?21?31?41?51?61?71?«1?913013113213313413513S1371.1113916014114214314414514614714B149ISO1511121*31541551561571581591601MIf?1631641651661*716S169170171172
oI
SUBROUTINE CALFOR 74/74 O"T=1 4.6*4.13 71-00-18 15.nl.Sfl PAGE
inn
ion
?oo
C CALCMLATF THF AUTO POWFP ÇPECTRl'K OF FORCF.C
«ULT3(<jCALAR<O.'5.O.P> .OFOPCF. .FSTAR) )
CALFOP
C
c
ccc
OUTPUT THf AUTO POWFR SPECTRl!" OF
4P
45
44
POINT 4flFOPMAT(lHl)00 44 JsltNNPRINT 4<!,JFORMAT(47H &MTO POWFR SPECTRUMjnilMMYsnilTPUT<SHPRnw(FPOWF.R,J) iCONTINUE
FORCE.
OF FOPCF AT1
CALCULATF THF MFAN SOIIAPF FORCE AT THE PUNPLF
FNO OF PUNOl E
ENPS.
MSF0R=ASSIGN(7FP0(NN.l))DO 5 K=1,MMSFOR=ASSIGNC(AnO(nELFTE(MSFOR),SURCOL(FPOWFR.K)))
5 CONTINUEJO1IMMY=OIITPUTL (4?H RESOLVED MEAN SOUARE FORCE AT RIINDLF F.N0S,5,
. PAGF(M«;FOR) )CC EVALUATE THF K.M.S. VALUES OF THE RESOLVED FORCES.C
PMSFOP=P0WF.R(MSF0P, ( 0 . 5 , 0 . 0 ) )JPIIMMY=OUTPUTL(3'»H ROOT MEAN SQUARE FORCES AT P.UNOLF FNDStS,
, PAGEIHMSFOR) )CALL OELFTFX(FORGFN,n)CALL CLEANUPBO TO in
6 CONTINUFCccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc cC CASE ? CC CALCI'LATION OF CPOSS POWER SPECTRAL OFNSITY OF FORCE.CC CC C C C C C C C C C C C C C C C C C C L C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C
cCALL REAni(fHpISPOW,0ISPOW)IF(ROTPIN)niSPOWsASSIGNC(OELRC(DELETE(0ISPOW),NN,NN))IF (MIDPIN)OISPOWaASSlGNC(OELPC(DELETE IDISPOÏOtJP.JP))
CC CALCULATE THE CROSS POvFP SPECTRUM OF PFSOLVED FORCE.C
TÇTARTxTRANS(CftNJ(TRfcMAT))PS0F0R*ASSIGNCJ|MIILT3< INVERT (TRAMAT) ,DISPOW, INVERT (TSTART) ) )IF<L0GTYP,E0.3tG0 TO 7PRINT Al
61 FORMATtlHl)PRINT 6?.!SPEC
6? FORMAT(fpflH CROSS POWER SPECTRAL DENSITY OF FORCF AT SPFCTRAL LINE.NO. ,12)IF(BOTP1N)PRINT 35
CALFORCALFORCOLFORCALFORCALFORCALFORCALFOPCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFOPCALFOR.CALFOPCALFORCALFORCALFORCALFORCALFORCALFORCALFORCHANGECHANGECALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALF ORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALFORCALfORCALFORCALFCJ»CALFORCALFORCALFORCALFORCALFOR
17317417«j17617717H1791*01*11«21B31P41«51P61P7lBfl1P910010110210310*1951O6107
îoa1O9?no
?08?09?10?11
OI
??•
«ÎMRROUTINE CALFOB 74/74 OpT=l R0UN0=*-»/ FTN
?35
?4O
?*5
?TO
275
CALLCONTINU"7
( L ( ( T ) ,7FRO(1,1) ) )IF (MIOPIM) inilMMV=v)OINV (SOP«AT (AOTOPF, 1 , 1 < JP-1 , 1 ) tZFPO< 1 , 1 ) )IF(MIOPru>«'ITOPF=ASSIGNC(JOlNV(tC)UMMY,ÇUHMAT(OF-LfTF(AllTOPF) , J P .
1 .MN-JP.1MI) ,RANWID>)
) .AIITOPF)
rF(ISPfC.GT.M)SO TO BGO TO 1
3 CONTINl'F.FnReow=AÇSlGNC(OELCOL(OELeTE(FOPPOW),l))
OUTPUT THF AUTO POWF& SPECTRA OF APPLIFD FORCE*.
PRTMT Hl01
no a? L=J.NNPRINT 83.L
83 FOPMAT(*1H AUTO POWER SPECTRAL OENSITY OF APPLTFO FOPCFS AT END OF.BUNDLE ,1?)jnilUMY=nuTPI)T'(S<IBPOU(FORP0W<L>)
B2 CONTINUF
CALCULATE THE "FAM SOUABE VALUES OF FOPCE.
PO 9 I J = I f MMSFOR=A15';lGNC(Ann(f1FLFTE(M^FOR) .SURCOL (FORPOW. IJ» ) )CONTINU?MSFORsASSIfiMCiÇMijLT (DELETE (MSFOP).BANWin))JpiJMMYsOIITPUTL |4?H M£AN SQUARE RESOLVEO FOPCf AT PUNPLF FNDSt5,
FVALHATE THE POOT "EAN SQUARE VALUES OF RESPONSE.
RMSFOR=POWE»(MSFOP.(0.5.0.0))JtlUMMY=OUTPi.lTL(47H ROOT MF>N SOUAPE RESOLVED FORCE AT BUNOLE ENDS.
. 6,PAGE(RMSFOR))CALL OELFTEX(FORGEN.O)CALL CLEANUP
Î0 CONTINUFCALL SMOWMEMRFTURNEND
-09-m
CALFOPCALFORCAL^OPC Al CAD
CALFOPCALFORCALFORCALFORCALFORCALFOPCALFORCALFORCALFOPCALFORCALFORCALFOPCALFOPCALFORCALFORCALf-OPCALFOPCALFORCALFORCAt FORCALFORCALFORCALFORCALFOP.CALFORCALFORCALFORCALFORCALFORCALFORC Ûi F OP
CALFOPCALFORCAlFORCAl.FORCALFOPCALFOPCALFORCALFORCALFORCALFORCHANGECALFOBCALFORCALFOR
15.01.58
??«??9?30
?31?3??33?3*?7S?36?37?38?39?40?4l?*2?43?*•?45?4A?47
?49?50?=U?S?rrij?«i4?S5?56?*7?S8?S9?t,0
?f>\
?*2?A3?6*?«.5?ft6?*7Pfifl?ft9?70?72?733
?74?75?76
PAGE
-212-
APPENDIX II
MATRIX IDENTITIES AND OPERATIONS
-213-
APPENDIX II - MATRIX IDENTITIES AND OPERATIONS
1. If (i) <f> and iC are column vectors,
(il) a is a row vector of ones,
(iii) n is a scalar matrix (order (lxl)),
and (iv) A is a square matrix
then: -
= A*<J) «
T T T T) Tï
f> * A = A*<|) J A <j) ît <J) A t
X*A = A*X = XA = AX
T T T D Dt <J> *A = A*<J> = A ()) ^ 0 A i f m a t r i x s y m m e t r i c
-T —2 . I f f ( x 1 , x 2 , x ^ , y l f y 2 , . . . . . . y ) - y . A • x
3ft h e n ~37 = 4 ' x
3f .-T ANT .T -and = (y . A) = A .y
3x ^ ^
-T -(y .A.x i s c a l l e d a b i l i n e a r t e rm. )
~T —A l s o , i f f ( x - , x , x ) = x . ^ . x ,
-214-
then, using the product rule for differentiation,
3f T -= A. x + A . x
3x
If A is a symmetric matrix then
3f2 *
-T -(x . A . x is called a quadratic term.)
-215-
APPENDIX III
STRUCTURAL DAMPING :
AN ALTERNATIVE IMPLEMENTATION
-216-
APPENDIX III - STRUCTURAL DAMPING: AN ALTERNATIVE IMPLEMENTATION
In section 2.5 it was assumed that each idealized fuel bundle has
three associa-ted structural damping factors (G , G , and G ).
This approach derives directly from the mathematical theory of
structural damping forces. The points of action of the equivalent
"structural dampers" can be seen in Figure AIII.l. Generation of
the structural damping factor vectors is described in AECL Report
6068, "DYNMOD: A Users Manual", to be published by V.A. Mason.
An alternative method for the implementation of structural damping
forces, in which new structural damping factors (G s, G _s and^ C S1 EN u
G s) are defined, is available for use with DYNMOD. Here, eachL AK.
e l a s t i c element along the fuel assembly has only one a s soc i a t ed
damping f a c t o r . This s i m p l i f i e d system of s t r u c t u r a l dampers can
be seen in Figure A I I I . 2 . Using t h i s i d e a l i z a t i o n , the s t r u c t u r a l
damping moment c o e f f i c i e n t mat r ices w i l l take the form:
1 ( G Î K 1 " 1 G 2 K 2.0
- i G 2 K 2 i (G2K2 - 1 G 3 K 3
G*K.)4 4
where (GT?KT) = (GCST^, KCST ) o r (GENDS, KEND ) ,J J J J J J
-217-
The equivalent matrix at present in DYNMOD is given by:
i* G l
G2
(
(
;3
" K
" K
" K
0.
~K
<F ' KN
where (G ,K ) = GCST , KCST ) or (GEND-, KEND-).J J J J J tj
The form of the above matrices is applicable to the basic fuel
assembly model with no additional constraints.
Both methods use the same 'bundle shear' structural damping
implementation, ie GPAR^ - GPAR .
DYNMOD can be converted to use the simplified structural damping
method as follows:
1. Change subroutine PINNED to the form shown in Figure AIII.3.
The scalar matrices DCBOT and DEBOT are the structural
damping factors associated with the end constraints KCBOT
and KEBOT respectively.
2. Replace statements DYNMOD 265 to 283 inclusive with the
statements shown in Figure AIII.4.
Adjust the CALL statement to suit the new subroutine PINNED,
and alter the integer declarations to define the new scalar
matrix pointers.
-218-
The scalar matrices DCTOPX and DETOPX in the main program are made
redundant by these changes. Numerical experiments with DYNMOD
using uniform beam idealizations show that identical vibration
characterist.ics are calculated for both methods of structural
damping implementation when (GCSTS = GC3T), (GENDS = GEND) and
(GPAR = GPAR).
-219-
STRUCTURAL DAMPING FACTORS FOR
BUNDLE .T-l ARE:
n C S T J _ 1 , GEND , nPAR
STRUCTURAL DA>fPINC FACTORS FORBUNDLE .1 ARE:
GCST , REND , OPAR
STRUCTURAL DAMl'JNC FACTORS FORBUNDLE .f+I ARE:
BUNDLE .1-1
BUNDLE J
BUNDLE .T+l
FIGURE A l l I . I : S t ruc tura l"Damners"Ideal ized bvDYNMOD.
STRUCTURAL DAMPING FACTORSFOR BUNDLE J - l ARE:
GCST S GENDST , GPAR^
STRUCTURAL DAMPING FACTORS FORBUNDLE .J ARE:
OCSTj, GENDT, GPAR,
STRUCTURAL DAMPING FACTORS FORBUNDLE .T+l ARE:
BUNDLE J - l
BUNDLE .1
BUNDLE .1+1
IGURE AIII.2: Sinrolified Structural
Damoing Idealization.
1U
3 t
<t0
b ' J j r O U T i l E H l N j - . £ C ; ( J F » ^ | N | f i i û o T t : i N ; ^ ï y P I ^ î ^ C 3 D T j K E B j T )iXC^CCu'-l. vLC C^L^J t ^TE THfc LCbr. DIM A TE HEDUC7ICW " A T * I X hG F i n A Pii.KEL. BUi-iuLt STRING.f l - h î fU tOCt - jT tFF+ t f câS -A M>- ^4-n-U G T IR HL D A-*4~Hvj -T-tfri-i^C ASSOCIATED WITH THE CONSTRAINED ENJ.GC
CGC
Cccccç€ -C GCGCGbùClGGCCGcCLCCGCCCGGCCCGCCCCoCGCi.CCCLGCCGCCCCCGCCCoGLGCGCCCGCCGCÛCCC LAhPS FUNCTIONS.C
ÎN lEGt r ALC,ASSiGNC,GGFY,CELET£,ÛlViSGALAR,. jKUi.1i,SUdfcfaTiSUdRJW.T^ANS,-kùiiPtf-x- -tfccHtfrT- •
CC LANPS SltR«GEC
INTEGfcr GGtuGr.PLEx SPACEGG.\;iCN / K 2 / G i h . / K 5 / h : OH/ tt<+/GUL/Kî>/LI N K / h b / SPACE
= û . . . L H L 3 L N L L E F K E E (ûcFâG'LT SETTING)= 1 . . . E N D JUN-ÛLE PINNED
- àt- i. «r-= - Û -!-»-._ : _ - . _ _ _ : _ : . : . _ .= J . . . E N D OF èUNCLt J FINNCC
KGDOT = ...£NL> GGN'iTf n l l . T AtulTIOt1* TC \/CSTKtcGT = . . .E i<D G U N Î T K M I I V T A C U I T I O N TÛ VéNCtXyOT = . . .DAMPING FACTCk FCk ÛONSTFAIMED END OKDE6GT = • • . C ArtP i N i i ffcLToK FCF CoNST-AINE G EuD f-LATE
L,CC
- • C -GGÇcCG
GGCGCGG
CC-C
C
c
S- AKG -tf^RI A ËLEs r— -
LOGICAL
SET DEFAOLÎ VALUES.
Al- , AZ,3tf,SZ,dPlN,CCdCT.0E3CT.i?
FlNNEùP
CCc
INPUT DATA
I » l .«•HJPIN, JP IN .EHKC8OT, KCBCT.
HKaOT KE3QT.
SHDEBOT, OESOT)tR£AL(ELEMEN7(e
PINNEDP1NNEÙ
IE:PiNNEÛ
r1NNEO•-P-iNwfS-PiNNtûPiNNEû
ûÇPiNNEtJPINNEDP j . K fn t UP-iNNEuP i ÔPINNEDPINNECPINNEDPISNEOPiNNEOPINNECPiNNEO
PINNEGPiNNEUPiNNtDFiNNECPINNEDFINNEDPINNEDPl^eFANNEDPiNNECPINNEDPINKEDPiNNEOPINNEDPINNEDP-1M.6-ÔPiNNEiJPINNED.PINNEDPINNEDPINNEOPINNEDPINNECP-irJvN-E-0PINNEDPINNEÛPINNEO
?
.1'•i
3à
ài
10111213
••-I-»-
1617la19
za21222«*
if282330-J-l-32333H353â3733M~-,1M2
<|ÔV7-Hd<i950= 1s2»3£4
FIGURE AIII .3 (part 1)
& U ÎC 3GO TL 1 >JJ p - NIF (MiLt -x I . ) JMreEAM CLEMENT U.~ i N . 1 , 1 ) )AN;= J G l N V ( I ^ e . \ ï ( J P - 1 ) t TK ANi (Nt G ( ù I \ / < 3031^41 (L ,
, - S U o t ^ M L , JP>> ) ) ) • --• -I F t 3u1r I i \ . J.NJ. .NGT.MIÛPÏN) Gu 7 0 3
( N - J F , J P - 1 )T ( P )
i F ( M l L F I h . i t . J . . - ' i C T . 3 0 T P l N ) GO 10 t>bZ=ZEKO(JP«N-JP-1)
70 i:-:= J û l f JdwEiJT ( N - J P - 1 ) , T S * N i ( N E G ( i J ;
c "s=AsIlGr.c(JCiriri < JCINVC hM.fiZ) . J L I I W (B/J.tM) ) )
GO T C ^ l o " "7b i ^ - ; b s > i G i » C (COPY ( « . - . ) )
7 C n t - T I N U fcC INTSOUUCt STRUCTURAL DAMPING ENù TEkMS.
• - • •£ — • - . . - - . — . ... . . .
60 KGrOT = <. i j lGNG(ACD(LE.L£Tt(KC.5UT)»SMLLT3(5CALf iR(0.• _ D C L ' C I «KuGOT ) ) )
0 , 1 . 0 ) ,
e, i . i ) ,
CALL ST
PiNNEOPINNEDPiNNECUNNEO-PIW.Euf* i. N N c P i NNEUPINNED
PiNNEUPINNEDPiN'Sto
PINIVEOPiNNECPINNEDPINNEDf i N K E CPINNEDPINNEDP1NNEOPlNNct:PINNEDPINNEUPINNEGPI^^EDPINNEDP1NNEO
clG 2c3i.-W-5es
1707172737H7576777d7 o-O-dld2s 3
a-»= 5
1IN»
NJ1
FIGURE A I I I . 3 ( p a r t 2 ) .
c • " "C If-TkOUUCE aTfcUCÏUaAl CAMPING TERMS.
KCSTti=«iii.IGN(SMULT3{SCALAR(0.'J.1.9),0CST,KC5T)> •- .Ktî-. CL,=b SS IGN(SMULT3 (SCALAR (0 . 0 , 1 . 0 ) » OENÛ ,<END > ) to0KFAku=ASSi&N(D(SMULT3(SCALAM 0.0 ,1 .0) .GPAK.KPAk)) ) NVll=»»iîSiiiNC,(ADD3(»/G.VCST,AOÙJ(VPARll,MULT3(fRANS<8ETA) ,0(<tSTû) . w
MSîIG A J D ( t f 2 1 , N E ( u K P A U ) ) )= ASSIl iM-(AC03(VEN(J, VPAR2i! , AGO (MLLT3 (TRANS ( BETA) ,O(KENCiO> .dETA)
0K»APC))>
FIGURE AIIT..4
-224-
APPENDIX IV
INTERBUNDLE PARAMETER GRADIENTS
- 225 -
APPENDIX IV - INTERBUNDLE PARAMETER GRADIENTS
The derivative (— I is calculated using the approximationsbelow. \ h
2 *-
for i = 1
" Z i Z i " for Ki<No + o t, +
i i+1 1-1
2 * for i = N
In matrix form these expressions become
dx
2000110001100020
1 0 0 0- 1 1 0 00 - 1 1 00 0 - 1 1
for a four bundle s t r ing .
In general,
Z T
1-10-1-00
0 -0
011 -0 -
00
l-l
1
dxY • (B r (-K). 1)
The function j r ~ is defined in statement DYNMOD 150 and the
constant matrices 3, K and y are constructed in statements
DYNMOD 239, 2 40 ancf 2^1 respectively.
-226-
APPENDIX V
LAMPS FUNCTIONS AND SUBROUTINES
USED IN DYNMOD
-227-
1. LAMPS FUNCTIONS USED IN DYNMOD
FUNCTION DESCRIPTION
ADD
ADD 3
ARGAND
ASSIGN
ASSIGNC
CONJ
COPY
CREATE
DECOL
Elemental or ordinary addition
(K=ADD(M,N) means jt = J +JJ)
Ordinary addition of three matrices
(K=ADD3(L,M,N) means J£ + L+]£+JJ)
An output function
(K=ARGAND(M) means plot an argand diagram ofthe complex elements of matrix M)
Protect a matrix
(K=ASSIGN(K) means protect matrix JC)
Protect a matrix and then erase a l l unprotectedmatrices in the computer core.
Evaluate the complex conjugate
(K=CONJ(M) means £ = J**)
Copy a matrix
(K=COPY(M) means that K i s a matrix whose elements
are ident ica l to JJ but J£ and JJ occupy different
portions of core)
Generate a LAMPS matrix from a complex array
(K=CREATE(C,JP,JQ,JPDIM) means create a (JPxJQ)matrix ^. which i s a copy of the complex matrix Cwhere C i s declared as a complex array and JPDIMis the dimensioned number of rows of C)
Convert a row or column vector into a diagonal matrix
(K=D(L) means that J£ i s a diagonal matrix whose i thelement on the diagonal i s the i th element of thevector matrix L or L
Delete a column of a matrix
(K=DELCOL(M,J) means that K equals M with column Jdeleted)
-228-
FUNCTION DESCRIPTION
DELETE Returns a matrix to the unprotected state
(K=DELETE(K) means that J£ is returned to theunprotected state)
DELRC D e l e t e a row and a column of a m a t r i x
(K=DELRC(M,I,J) means J£ e q u a l s M w i t h row Iand column J deleted)
DELROW Delete a row of a matrix
(K=DELROW(M,I) means J£ equals JJ with row I deleted)
DET Calculate the determinant of a matrix
(C=DET(M) means calculate the complex determinantC o f J!)
DIAG Convert a diagonal matrix into a column vector
(K=DIAG(M) means that K is a column vector con-sisting of the diagonal elements of M)
DN Elemental division
(K=DIV(M,N) means £
ELEMENT Return a matrix element
(C=ELEMENT(M,I,J) means that the complex number Cis equal to element (I,J) of M)
GETCOL Returns the number of columns of a matrix
(MCOL=-GETCOL(M) means that the integer MCOL isequal to the number of columns of matrix M)
GETROW Returns the number of rows of a matrix
(MROW=GETROW(M) means that the integer MROW isequal to the numb-er of rows of matrix M)
IDENT Generate an identity matrix
(K=IDENT(IP) means that matrix K has ones on thediagonal and zeros elsewhere; J£ is of order (IPxIP))
INVERT Calculate the inverse of a matrix
(K=INVERT(M) means JC=) ~1)
JOINH Join two matrices horizontally
(K=JOINH(M,N) means
- 2 2 9 -
FUNCTION DESCRIPTION
JOINV
LOWER
MODAL
MULT
MULT3
NEG
NORMLIZ
ONE
ONEBLO
ONEBOV
OUTPUT
Join two matrices ve r t i ca l ly
(K=JOINV(M,N) means £= ("))
Generate a matrix with ones on and below the maindiagonal and zeros elwewhere
(K=LOWER(IP) means form the (IPxIP) matrix £ asdefined above)
Solves the eigenvalue problem (M-X*^) .£=(}, where JJis a (nxn) square matrix.
(K=MODAL(M,IOUT) means that IÔÏÏT i s returned asa column vector of the eigenvalues X in ascendingorder of modulus. K is the normalized modal matrixof M such that the j th column of K i s the normalizedeigenvector X. corresponding to the j t n eigenvalue)
Matrix mult ipl icat ion
(K=MULT(M,N) means J^ .JJ)
Matrix mult ipl icat ion
(K=MULT3(L,M,N) means J^.Jg.JJ)
Elemental negation
(K-NEG(M) means £=-J!J)
Normalizes the columns of a matrix
(K=NORMLIZ(M) means that the elements of eachcolumn of JJ are divided by the magnitude of thecolumn; th is leaves the normalized-matrix Jp
Generate a matrix of ones
(K=ONE(IP,IQ) means £ i s an (IPxIQ) matrix of ones)
Generates a matrix with ones below the diagonal
(K=ONEBLO(IP) means that matrix £ has ones belowthe diagonal but zeros on and above i t )
Generates a matrix with ones above the diagonal
(K=ONEBOV(IP) means that matrix K has ones abovethe diagonal but zeros on and below i t )
An output function
(K=OUTPUT(K) means pr int on the lineprinter)
-230-
FUNCTION DESCRIPTION
OUTPUTC
OUTPUTL
OUTPUTT
PAGE
POWER
POWERN
SCALAR
SCALARC
SDIAGHI
SDIAGLO
An output function
(K=OUTPUTC(TITLE,J,K) means print J£ togetherwith J one-word t i t l e s , one above each of thef i r s t J columns (l£J£4))
An output function
(K=OUTPUTL(TITLE, J,K) means pring JC togetherwith a t i t l e consisting of J words (1<J<8))
An output function
(K=OUTPUTT(TITLE,K) means printa t i t l e of up to 10 characters)
together with
An output function
(K=OUTPUT(PAGE(M)) means skip to a new page onthe lineprinter and print M)
Elemental exponentiation
(K=P0WER(M,C) means £ is the matrixthe power C, K * C
raised to
Elemental exponentiation
(K=POWERN(M,N) means J£ is the matrix Jg raisedto the integer power N, i . e . each element of Jg israised to the power N)
Matrix generation
(K=SCALAR(R, AI) means that JC is a (lxl) matrixwhose real part is R and whose imaginary part isAI)
Matrix generation
(K=SCALARC(C) means that | is a (lxl) matrixcontaining the complex number C)
Generates a matrix with ones on the super-diagonal
(K=SDIAGHI(IP) means that £ has ones on thesuperdiagonal and zeros elsewhere; K is ofthe order (IPxIP))
Generates a matrix with ones on the subdiagonal
(K=SDIAGLO(IP) means that J£ has ones on the sub-diagonal and zeros elsewhere; K is of order(IPxIP))
- 2 3 1 -
FUNCTION DESCRIPTION
SIGNMOD Forms the moduli of the elements of a matrix
(K=SIGNMOD(M) means that the element K±j of Kis equal to the modulus of element M^J of JjJ.K.. has the sign of the real part of M )
SMULT
SMULT3
Elemental mul t ip l ica t ion
(K=SMULT(M,N) means J£ = J
Elemental mul t ip l ica t ion
(K=SMULT.3(L,M,N) means £
* JJ)
J. * JJ * JJ)
SPECIAL
SUB
SUBCOL
SHBMAT
SUBROW
TRANS
TRIINV.
Generates a matrix of zeros with unity added toone of the elements
(K=SPECIAL(M,N,J,K) means that K is a (MxN)matrix with unity in position (J,K) and zeroselsewhere)
Elemental or ordinary subtraction
(K=SUB(M,N) means K = JJ-JJ)
Generates a column vector from the column of amatrix
(K=SUBCOL(M,J) means that K is a column vectormatrix equal to column J of matrix M)
Generates a submatrix from a matrix
(K=SUBMAT(M,I,J,JP,JQ) means that £ is a (JPxJQ)submatrix of M. The element (1,1) of K corres-ponds to element (I,J) of M)
Generates a row vector from the row of a matrix.
(K=SUBROW(M,I) means that K is a row vectormatrix equal to row I of matrix M)
Form the transpose of a matrix
(K=TRANS(M) means £=fJT)
Calculates the inverse of a tridiagonal matrix
(K=TRIINV(M,FUZZ,LSW,NZ,10UT) means that K is theinverse of the tridiagonal matrix M. If B issingular then LSW is set equal to TRUE, idUT givesthe indices of the zero rows and NZ is the numberof rows in IOUT. All numbers £ FUZZ are regardedas zero)
ZERO Generates a matrix of zeros
(K=ZERO(IP,IQ) means thatof zeros)
is an (IPxIQ) matrix
-232-
2. LAMPS SUBROUTINES USED IN DYNMOD
SUBROUTINE DESCRIPTION
CLEANUP
DELETEX
GRAFM
INITIAL
READn
SHOWMEM
STORE
Erases a l l unprotected matr ices from core
(CALL CLEANUP)
Return matr ices to the unprotected s t a t e
(CALL DELETEX(M.N) means tha t a l l matr ices createdbetween fll and JJ are re turned to the unprotecteds ta te )
Parallelogram plot t ing routine
(CALL GRAFM(M,N,L,K1,K2,K3,K4,K5); for a ful ldescription see Report AECL-5977.
A subroutine used for matrix storage management;see iJeport AECL- 5977.
(CALL INITIAL(HISPACE,HIDIR))
Input subroutine
(CALL READn(6HTITLEl,Ml ôHTITLEn,Mn) meansthat the n matrices J£l, J n are input in freeformat. The matrix t i t l e s appear on the datacards with the matrix data)
Indicates the amount of core space required bythe program
(CALL SHOWMEM prints out the highest locationreached in the LAMPS matrix storage array SPACEsince the s t a r t of the job)
Changes a matrix by inser t ing a complex number
(CALL STORE(M,I,J,C) inser ts a complex number Cin the element location ( I , J ) of matrix M)
-233-
TABLE 1
DYHMOD INPUT PARAMETERS AND VARIABLES
MATRIX ELEMENT UNITS MATRIXVARIABLE DIMEN- ORDERNAME SIONS *
DATA INPUTSET OPTIONNUMBER
MATRIXDESCRIPTION
-1HzBANWID T
BPIN
BWR
CDCFCB I-T"1, m/s (Nx3)
DATA
DBUG
DCBOTX
DCTOPX
DEBOTX
DEQ L
DETOPX
,-2DFORCE
DIAM
DIAMH
MLT
L
L
N
m
m
(lxl)
(lxl)
(lxl)
6,7,9
4
1
Frequency interval
End bundle constraintindicator
Plotter bundle length/width ratio
(1x1)
(lxl)
(lxl)
7
1
4
(lxl)
(lxl)
(Nxl)
(lxl)
(Nxm)
(Nxl)
(Nxl)
Horizontally joined coef-ficient vectors of viscousdrag (CD), friction (CF) andbase drag (CB)
Force input descriptor
Program debugging indicator.
Extra interbundle bending stiff-ness structural damping factorat bottom constraint
Extra interbundle bending stiff-ness structural damping factorat top constraint
Extra endplate flexure stiff-ness structural damping factorat bottom constraint
Equivalent diameter of the freeend.
Extra endplate flexure stiff-ness structural damping factorat top constraint
Discrete spectra of appliedforces
Fuel element diameters
Bundle/pressure tube hydraulicdiameters
(cont'd)
-234-
MATRIX ELEMENT UNITS MATRIXVARIABLE DIMEN- ORDERNAME SIONS *
DATA INPUTSET OPTIONNUMBER
MATRIXDESCRIPTION
DISMAT
DISPOW L T
EXPER
FEF
MAX
FMIN
FORCE
G
GCST
GENT)
GPAR
JPIN
KCBOT
KCST
KEBOT
KEND
KPAR
-1
-1
MTL-2
LT,-2
m
Hz
Hz
m/s
(Nxm)
m /Hz (NxN)
(lxl)
(Nxl)
(lxl)
(lxl)
(Nxl)
(Nxl)
2 - 2ML T l
2 - 2
2 - 2ML T
ML2T"2
MLV 2
N.n/rad
N.m/rad
N.m/rad
N.m/rad
N.m/rad
(lxl)
(lxl)
(Nxl)
(lxl)
(Nxl)
(Nxl)
11
11
3
6
6,7,9
(Nxl)
(lxl)
(nxl)
5
2
2
Discrete spectra of dis-placement response
Cross-nower spectral densityof displacement response
Transfer function datadescriptor
Free end shape factor
Upper frequency limit oftransfer function
Lower frequency limit ofspectra
Applied harmonic forces
Gravitational acceleration
Structural damping factorsassociated with interbundlebending stiffnesses
Structural damping factorsassociated with bundle end-plate flexure stiffness
Structural damping factorsassociated with bundle shear
Intermediate bundle constraintindicator
Interbundle bending stiffnessat bottom constraint
Interbundle bending stiffness
Endplate flexure stiffnessat bottom constraint
Endplate flexure stiffnesses
Bundle shear stiffnesses
•(cont'd)
- 2 3 5 -
MATRIXVARIABLENAME
L
M
NEL
ELEMENTDIMEN-SIONS
L
M
-
UNITS
m
kg
-
MATRIXORDER
*
(Nxl)
(Nxl)
(Nxl)
DATASETNUMBER
2
2
2
INPUTOPTION
-
-
-
MATRIXDESCRIPTION
Bundle lengths
Bundle masses
Numbers of fuel elementsper bundle
NPERFR
OMEGA
PLOT
POINTS
PSDFOR
RHO
SAMPLE
SDR
SHAKER
TMAX
TRAMAT
TYPE
U
ML
T
- 3
- (lxl) 1 - Number of modeshapes plottedper frame
Hz (lxl) 5 - Frequency of the applied
harmonic forces
(lxl) 1 - Graph plot indicator
(lxl) 7,9 - Number of frequency pointsof interest in a calculation
M 2 L 2 T " 3 S 2 / H Z ( N X N ) 8
kg/m
s
-
(Nxl)
(lxl)
(lxl)
(lxl)
3
5
1
6
UNCORF M 2L 2T~ 3 N 2 /Hz (Nxm)
VM
Averaged cross-power spectraldensity of the applied forces
Fluid densities
Interval between time-stepcalculations
Modeshane plot successivedeflection ratio
Number of bundle to whichthe electrodynamic shaker isattached
s (lxl) 5 - Transient response observation
time
n/N (NxN) 10 optional Cross receptance matrix
- (lxl) 9 - Response input data descriptor
for resolved force calculations
3 - Flow velocities
8 3 Uncorrelated applied forces
kg (Nxl) 3 - Hydrodynamic (virtual oradded) masses along the fuelstring
m/s (Nxl)
(cont'd)
-236-
MATRIX ELEMENT UNITS MATRIXVARIABLE DIMEN- ORDERNAME SIONS *
DATA INPUTSET OPTIONNUMBER
MATRIXDESCRIPTION
WHICH
XL
YL
(lxl)
in. (lxl)
(lxl)
in. (lxl)
Harmonic forced responsecalculation indicator
Plot frame length in Xdirection
Gravitational accelerationdirection indicator
Plot frame length in Ydirection
* N is the number of bundles in the fuel string.
m is the number of spectral lines in frequency domain.
-237-
TABLE 2
MATRIX OPERATORS
SYMBOL OPERATION
31
3x
-1
Column vector,
Column vector,
Row vector,
Rectangular matrix
Elemental addition
Elemental subtraction
Elemental and scalar multiplication
Elemental and scalar division
Matrix multiplication
Inverse of matrix ^
Transpose of matrix ^
Diagonal matrix form of the vector JÏ
ISSN 0067 - 0367
To identify individual documents in the serieswe have added an AECL- number to each.
Please refer to the AECL- number when re-questing additional copies of this document
from
Scientific Document Distribution Office
Atomic Energy of Canada Limited
Chalk River. Ontario. Canada
KOJ 1J0
ISSN 0067 - 0367
Pour identifier les rapports individuels faisantpartie de cette série nous leur avons ajoutéun numéro AECL-.
Veuillez faire mention du numéro AECL- sivous demandez d'autres exemplaires de cerapport
Service de Distribution des Documents Officiels
L'Energie Atomique du Canada Limitée
Chalk River. Ontario. Canada
KOJ 1J0
Price $11.00 per copy Prix $11.00 par exemplaire
2374-78