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


7.2 Computation 78

8. TRANSFER FUNCTIONS 79


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, 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.


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-


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


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.


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-


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.


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 = ((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


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.


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

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, - .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


IMAG,

LIMIT A,CD = 0.0 m/sLIMIT B.CD = 0.3 m/s

VARY CF

NOMINAL VALUE -0.022


IMAG,

LIMIT A,CF = -0.01LIMIT B,CF = -0.03

VARY VH

NOMINAL VALUE 10.17 kg(VM = HYDRODYNAMIC MASS PER

BUNDLE)


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

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

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74/74 OPT=1 ROMND=*-*/ FTN 4.6*433 7R-09-1B 15.01,58 PAGE

70

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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

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PAGF

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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

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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-


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 -


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-


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 -


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-



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-



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

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