SEPARATION CRITERIA OF FLUID PIPING SYSTEMS USING A LUMPED-PARAMETER METHOD

by

Rajesh Rungîa

#.

Submitted in partial fulfilment of the requirements for the degree of

Master of Engineering Science

Faculty of Graduate Studies The University of Western Ontario

London, Chtario March 1998

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ABSTRACT

This thesis uses the similarity between the dynamics of structural systems and acoustic synems to show that structural uncoupling aiteria are applicable to acoustic systems. In the analysis uncoupIing criteria is developed using two degree of tieedom lumped parameter structural systems. The uncoupling criteria was applied to a continuous acoustic syaem, where the continuous system was replaced with an equivalent lumped system for the first mode. Shifts of fiequencies were estimated using the uncoupling criteria. The application of lumped parameter uncoupling criteria was vedïed by using finite element software-computer generated runs of some exarnples.

In the specific uncoupling criteria we limit the shift of primary syaem fiequency to ten percent due to attachment ofa secondary system. In an unrestrained secondary syaem, the SM is dependent on the fkquency ratio and the mass ratio of secondary to primary system masses. Higher shiHs are obtained as the fiequency ratio approaches one, or as the mass ratio increases. In a restrained secondary system, the fiequency shift of the primary system, in addition to the above two factors, is dependent on the static remaint of the secondary syaem. The analysis shows that the frequemcy shift of the primary system is greater for an unrestrained system than b r a restrained system, using approximate equivalent parameters.

The criteria developed was used to assess the shifts in continuous systems. In order to apply the theory, a continuous system was idealized using lumped parameter systems. For developing the two degree of fieedom syaem, apprciximate displacement shape of the continuous system was used. The criteria therefore gives approximate shifts and are only applicable for assessrnent purposes. Cornparison of an acoustic system with a structural system shows that these equilibrium equations in these two SyRemS are exady the same. ThereTore, the uncouplhg cnteria developed for structural systems are applicable for acoustic synems.To apply the uncoupling criteria to acoustic systems, continuous lumped parameters similar to those obtained for structural systems were developed. The appiicability of these parameters were assessed numerically using two examples.

iv

ACKNOWLEDGEMENTS

1 express my greatest appreciation to Dr. Base for his supervision of this research project.

1 also would like to thank AMnd Misra, and Dr .D.K. Vijay of Ontario Hydro, for their technical support of this project .

Finally, 1 am gratefbi to my parents, Nishi Dhir, and Preeti Rungta for their moral suppon and encouragement.

TABLE OF CONTENTS

CERTIFICATE OF EXAMINATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii ACKNOWLEDGEMEIWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v, vi LISTOFFIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii ... LISTOFTABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vui

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

CHAPTER 1 1.0 INTRODUCTION

1.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Uncoupling of Piping Systems 3

. . . . . . . . . . . . . . 1.3 The Significance of Analyshg Structural Piping Vibration 4 1.4 Acoustic Piping Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

. . . . . . . . . . . . 1 -5 Justification For Mathematical Separation Of Piping Systems 9 1.6 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

CHAPTER 2 2.0 THE STRUCTURAL MODEL (LUMPEPMASS OR SPRINC MASS SYSTEM)

2.0 Eigenvalue Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1 Non-dirnensionalization o f frequency parameters . . . . . . . . . . . . . . . . . . . . 12 2.2 Two degree of fkeedom unrestrained system . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Discussions for the unrestraineû secondary system . . . . . . . . . . . . . . . . . . . 26 2.4 Two degree of &dom syaem with restraint . . . . . . . . . . . . . . . . . . . . . . . 28 2.5 Discussion for the restrained syaem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6 Discussion of Results for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

CEIAPTER 3 3.0 THE ACOUSTIC MODEL (CONTMUOUS SYSTEM MODEL)

. 3 0 Acoustic Resonance Phenornena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 3.1 Dariington Nuclear Generating Station Acoustic Vibration Problem . . . . A . 37 3.2 Effect of Transmission Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 Effect of Viscous Flow in Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3 -4 EEect of Pipe Row . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.5 Effécts of Diameter of Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.6 Attenuation (for pressure pulses in the system) . . . . . . . . . . . . . . . . . . . . . . 43 3 -7 Reflection (Compressive waves) ................................ -44 3-8 Axial Modes vs . Radial Modes for the fluid . . . . . . . . . . . . . . . . . . . . . . . . 44

CHAPTER 4 4.0 CONTINUOUS SYSTEMS

4.0 Continuous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46 4.1 Mathematical Ideaikation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Approximation o f Continuous System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 Lumped-Parameter Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.4 Demonstration of Sirnilarity behuem lumped-parameter and contuiuous

systems....................................................49 4.5 F i t e Difference formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6 Analogy between acoustic and structural systems . . . . . . . . . . . . . . . . . . . . 55 4.7 Equivalent mass and stifhess parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.8 Uncoupling Adysis of Acoustic Syaems by ABAQUS Runs . . . . . . . . . . . 62 4.9 Demonstration of Applicabüity of uncouplhg criteria to acoustic problems to

acoustic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.10 Discussion of Resuhs of ABAQUS examples . . . . . . . . . . . . . . . . . . . . . . . 72

CFiAPTER 5 5.0 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

REFERENCES 75

APPENDIX A: ILLUSTRATIONS OF PIPING SYSTEMS 78 APPENDIX B: ABAQUS INPUT DECKS AND OUTPUT DECKS ai APPENDIX C: DEIELOPMENT OF EQUATIONS tQ4

APPENDIX D: COAMPUTER PROGR4M FOR NUMERICAL 1 Ir CALCULATION

121

vii

LIST OF FIGURES

Fig 1-4 Fig 5 Fig 6 Fig 7 Fig 8 Fig 9 Fig 10 Fig 1 1 Fig 12 Fig 13 Fig 14 Fig 15 Fig 16 Fig 17 Fig 18

Exampie of pnmary and secondary piping systems . . . . . . . . . . . . . . . . . 5.6. 7 Picture of schematic drawing of primary and secondary piping systems . . . . . 8 SDOF Primary-Unrestrained SDOF-Secondary System . . . . . . . . . . . . . . . 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency shifl graph 18

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency shiA graph , . . . . . . . . . . . 18 Frequency shifl graph (mas ratio vs . theta) . . . . . . . . . . . . . . . . . . . . . . . . 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proposed Uncoupling Criteria 26 SDOF Primary-Restrained SDOF-Secondary System . . . . . . . . . . . . . . . 29 Dynamically Equivalent Primary-Unrestrained Secondary System . . . . . . 31

. . . . . . . . . . . . . . . . . Pressure Standing Wave Of Organ Pipe Resonance 38 Heat Transport System-Typical Feeder and Header Arrangement . . . 44 37-Element Fuel Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Element of rod 57 Diagram of unrestrained acoustic system . . . . . . . . . . . . . . . . . . . . 68 Diagram of acoustic restrained system . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Table 1

Tahle 2

Table 3

viii

LIST OF TABLES

Equivalency table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1

Petcentage error table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Sensitivity study table . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . 70

LIST OF SYMBOLS

area

speed of sound

diameter

modulus of elasticity

shift in frequency

sphg constant or stifiess

spring constant of primary mass

spring constant o f secondary system

second spring constant of secondary system

lengt h

pressure

displacement

natural fiequency for primary system ( uncoupled )

natural frequency of the secondary systern ( uncoupled )

shift m the coupled fiequency of the primary system

shift in the coupied hequency of the secondaiy system

density

mode shape factor

mode shape factor

strain energy

mess

fiequency of the primary system ( coupled )

fiequency of the secondary system (coupled )

wavelength

mass ratio of the secondary system to the primary system

natural frequency

INTRODUCTION

1.1 GENERAL REMARKS

This thesis will cover the separation criteria of the vibration of fluid piping

systems using a lumped parameter method to mode1 an analagous acoustic system. The

first part of the thesis will discuss the uncoupling criteria of piping systems. When

describing the uncoupled analysis of piping systems, an analagous structural system will be

presented. It was shown that the dynamics of the acoustic system and the structural

systern are very similar. The goal of this research study is to provide the uncoupling

critena. A method will be provided, so that any engineers solving complex problems with

pipe networks can determine whether to analyse the pnmary and secondary system

together or separately. With the uncoupling criteria provided, engineers will be able to

analyse complex systems separately, minlmizîng effort, and saving mon-. The objective of

this nudy is not to provide acouaic analysis, or compute response calculations. This

thesis addresses the question of whether a complex system can be analysed separately or

not .

This project stemrned fiom the niel failure (breakage of the endplate on the fût$

bundle) problem in the fiel channels at Darlington Nuclear Station [43]. The pressure

pulsations that were generated by the pump were c h e d through the header in to the

individual feeders, and caused the breakage of the endplate on the fuel bundle.

When the endplate breaks on the fiel bundle, the assembly of 37 fuel elements becomes

2

unstable, and the fuel itself fiactures. When the fbel fiactures, it goes into the primary heat

transpon system (PHT system) and then there are problems of radioactivity. The result is

a so-called "Nuclear Accident ".

A complete analysis of this problem is highly complicated. By using the

uncoupling criteria developed in this thesis (for the spring-mass system in chapter 2), it is

easier to analyse this acoustic vibration problem. Although, the acoustic vibration problem

has to be first modelled using a lurnped-parameter approximation. If the fumped

parameters developed are correct, then the frequency shifi of the primary system can be

calculated using the uncoupling criteria. The specific problem that this thesis will address

is to find out what is the frequency shift of the pnmary system by doing an uncoupied

analysis. Hence, it will be considered that if the critical frequency shiA (10%) is reached

or not, by a uncoupied analysis If the cntical frequency shift is above 1096, then a coupled

analysis will be required, which means a lot of manpower and resources. Othenvise, if the

critical shifl is less than 10% the various pipe networks may be analysed separately. In

short, a coupled analysis is what is trying to be avoided by doing a prelimlnary uncoupled

analysis first .

1.2 UNCOUPLING OF PIPING SYSTEMS

Piping systems can be thought of as large systems consiaing of the main system, a

large diameter pipe, and the secondary system, a smdler diameter pipe, comected to the

main pipe. A typical example of a pipuig system is one piping system (4" pipe) and two

secondary systems (1 /2" pipe).

The procedure of decomposing a whole system down into two or more systems is

called uncoupiing. It is done to find out how diRerent characteristics of the main syaems

change when a small system (in cornparison) is attached to it. The syaems are represented

in terms of mass and n i f iess

Therefore dynamic analysis of a complex systern cm be simplifieci by breaking

donn the syaem to primary and secondary systems. Each syaern can then be analysed

independently with the response of the primary systern acting as input to the secondary

system .

One of the critena consists of checking if the addition of a secondary system

sipnificantly modifies the primary qstem natural fkequency. In this report, the shifi of the

pnmary system fiequency due to anachment of a restrained secondary systern is -

analysed After that, this analogy is used to analyse the vibration of an acoustic systern?

4

1.3 THE SIGNIFICANCE OF ANALYSING STRUCTURAL PIPING

VIBRATION

Piping systems are one of the most important systems of today's industrial and

power generation facilities. It is vital that the fluids they cany be transmitted uninterrupted

for safe operation of a facility. Therefore, much care is taken to analyse these systems to

obtain the design forces that are a part of them [44].

Some piping systems cover large areas of a facility, transponing fluids to places

that are at different elevations and far apan. The piping syaems are provided with several

suppons between their ends. It c m be found that often a designer has to carefully balance

the two confiicting requirernents when choosing the supports in a piping system. The loads

caused by the operating temperature necessitate the piping to be as flexible as possible to

avoid large stresses due to thermal expansion [44]. Therefore, it is more desirable to have

a fewer number of suppons that allow thermal expansion. But, a piping syaem should also

be adequately supponed to reduce large bending deformations caused by a building

vibration during an emhquake.

The main structure causes the piping supports to move, during an eanhquake. X

These piping support movements are not unifoxm and Vary greatly. At various points they

could be in and out of phase with respect to each other, or the intensity with which they

move, could differ a lot. But, though the piping support movements m different, they are

correlated with each other. Thîs is because they are caused by the vibr'ation of the

supponing aructure that is subjected to a ground motion at its base.

5

Figures 1 . 2 , 3 , and 4 show piping systems (primary systems and secondary systems) at

Darlington Nuclear Station.

Figure 5 is a schematic drawing of an actuai piping system at Darlington Nuclear Station.

The purpose of figures 1-5 are to show actual pictures of the piping systems in a plant.

Fig.1 Piping Systom at Darlington Nuclear Station (primary p i p i n g is the big h o r i z o n t a l p i p e and the socmdary system is the vertical pipe)

Fig. 2

Figure 5 -Schematic diagram showhg pr- md wcondary piphg system~ at Dulington Nuclcar Station.

1.4 ACOUSTIC PIPING VIBRATION

A pump initiates fluid vibration by causing dynamic pressure pulsations. These

dynamic pressure pulses are arnplified by the fluid medium (acoustic medium) and these

amplified pressures (forces) could excite piping motion.

The piping vibrations are caused by fluid forces at area discontinuities e.g. elbows

and branch connections. These are due to the dynamics of the fluid contained within the

piping system and a dynamic pressure source such as a pump or valve [45]. The scope of

this section is to study the dynamic behaviour of a piping system due to fluid forces. The

emphasis is on analysing a small pipe co~ec ted to a large pipe. This topic will be covered

in chapter 3.

1.5 JUSTlFICATION FOR UNCOUPLINC OF PIPING SYSTEMS

When engineers analyse any large qaem at one time, it is caiied a coupled

analysis; and it can be very complicated. Sometimes, depending on particular

characteristics of a system, it is too complicated; and it requires too many resources, too X

much time. and too much money. It is for these reasons that uncoupling of systems is

preferred for many analyses.

Sometimes secondary systems are designed and constructed after the primary

system has been designed and constructed. Therefore some characteristics of the whole

-stem change, afler the secondary system is attached to it. These new changes in the

10

characteristics of the system can sornetimes be significant, and potentially dangerous. For

exarnple, if there is a large diameter pipe (which is the main system), and there are smaller

diameter pipes attached to it, the smaiier size pipes might significantly alter the fiequency

at which the main pipe vibrates (in case of an emhquake). If the main pipe happens to be a

pressurized pipe, this change in fiequency, depending on the location and application of

the pipe, could be extremely hazardous. Thus, one of the reasons why uncoupling

analysis is done is to find out if the secondary system can cause any major dmerences in

the main system.

Cncoupling analysis is very useful for this problem. By doing an analysis, it can be

found, for example, for what criteria is the secondary system going to change the

fiequency at which the primary syaem vibrates. It might be found that if the equivalent

mass ratio (for the acoustic vibration system) is below a certain ratio, only then should

the secondary mass be taken into consideration. Or it may be determined that if the

stiffhess ratio between the primary system and secondary system is below a certain ratio,

only then should the secondary system be considered. These conclusions can be reached

for both structural systems and acounic systems.

f .6 PREVIOUS WORK

There is some information in the literature on the uncoupling critena for restrained

secondary systems subjected to seismic input. However, there is no information on

separation cnteria for acoustic systems.

Pickel [25] suggeaed general guidelines for the maximum mass ratio for

uncoupling to be reasonable for each frequency ratio. N.M.Newmark, J.A. Blume, and

K.K. Kapur [24] studied seismic design criteria for nuclear power plants. Crandall and

Mark [12] researched the eflects of mass ratios on the mean-square response to white

noise Gupta [5] examined seismic malysis and design of structures and equipment such

as those found in nuclear power plants and petroleum facilities.

A. K. Gupta and J.M. Ternbulkar studied simple methods of combining modal

responses A.H.Hadjian [2] also researched on a similar topic by looking at seisrnic

response of structures by using a response spectrum method. Other authors who

examined modal analysis techniques was A.J.Salmonte 131, dong with D.W. Lindley and

J.R. Yow [14] who looked at modal response summations for seismic qualification.

CHAPTER 2

2.0 EIGENVALUE ANALYSIS

The starting point for any audy in uncoupling analysis is the two-degree of

freedom system for a simple mass spring system.It should be thought of as a modal two

degree of freedorn system representing any two modes; one for the primary system and

one for the secondary sy stemThe characteristic parameters for this two degree of fieedom

system are, the hequency of the pnmary system, o, , or alternatively one of its modal

fiequencies; the frequency of the secondary system, o, or ahematively one of its modal

fiequencies; and the mass ratio, p, which is defined as the ratio of the modal mass of the

secondary system to the modal m a s ofthe pnmary system. The modal mass ratio is the

same as the actual mass ratio, if the primary and the secondary synems are both

considered one degree of fieedom systems. This is not generally true for multi-degree of

freedom systems.

2.1 NON-DIMENSIONALMATION OF FREQUEKCY

PARAMETERS

X

In order to understand the problems that caused the breakage of the endplate on

the fuel bundle (as discussed in the "General Remarks 1.1 ") , it is imponant to understand

the intrinsic relationship between the primary syaem and the secondary system. In the

foiiowing anaiysis the author of this thesis non-dimensionalizes the masses of both

13

synems, the stifhesses of both systems, and moa importantly of dl, the frequencies of

both systems are non-dimensionalized. By non-dimensionalizing these respective

quantities, a conclusion c m be made for the structural system which will be used for

further analysis of acoustic systems. This method is a medium so that engineers can

compare sirnilar entities, and have discussions on the behaviour of complex mechanism.

2.1.1 THEORY OF DIMENSIONAL ANALYSIS

Dimensional analysis is a method for reducing the number and complexity of

variables which affect a given physical phenornenon. If a phenornenon depends upon n

dimensional variables, dimensional analysis will reduce the problem to only k

dimensionless variables, where the reduction n-k=1,i, 3. or 4. depends upon the problem

complexity 1461. Although its purpose is to reduce variables and group them in

dimensionless forrn, dimensionai analysis has several side benefits. The biggest benefit is

an enormous saving in time and money when analyzing rnany systems.

2.2 TWO DECREE OF FREEDOM UNRESTRAINED SYSTEM

Fioure A

Fig 6-unrestrained secondary system

The natural fiequencies for the coupled system are given by:

(K is the stifiess, o is the frequency, and hl is the mars).

fiom which

The natural fkquencies F, and 6 for the wupled system are rclated to the

uncoupled systems natural fiequencies o, and o, by the foiiowing relationships: ( note

that is defined to be the higher naturd fiequency whileq is defined to be the iower).

and

By non-dimensionalizing al1 of the parameters, a cornparison will be made later in the

thesis between the unrestrained system and the restrained system. The objective of the

following argument is to get al1 the significant parameters into get~eraiized no»-

dmr , rs~oiinlized ratios (mcrss ratios, andfieqt~mcy ratios). Wit h these non-

drmrrdo»iaked ruaos, we will be able to better understand the problem.

defining.

The following equations can be developed:

f; +f'= 1+ (1 + p ) 8 (2.28) -

The following expressions result by solving the above equations. The significance of the

two followiny equations will be shown in figures 7 and 8.

It cm be observed that the coupled system exhibits two frequencies which are

separated more than the uncoupled frequencies. One of these fieguencies is always above

the two uncoupled frequencies and one is always below both of the two uncoupled

frequencies. This push-nght', push-lefit phenornenon is depicted in Figure 7 and Figure 8.

(In the figure, "c" stands for coupled, "p" aands for pnmary, and "s" stands for

secondary)

Fig 7-fiequencies of pnmary, secondary, and coupled system

Fig. 8-fiequmcies of primUy,secondary, and coupled systm

Defining E, and e2 to represent shifts in the coupled fiequencies (eigenvalues of

the coupled system) from the square of the uncoupled wuencies (eigenvalues of the

uncoupled systems) to the right and left respectively show in Figure 7 and 8, the

following non-dimensional useful relationships can be derived (applicable for any mass

ratio and any fiequency ratio).

(On the following pages (22-25), the fiequency shifl equations will be non-

dimensionaiized, so that the uncoupling criteria can be graphed. A thorough explanation

will be given in the discussion and conclusions.)

Case of 8 r 1.0 t w- - r w. )

It can be show that:

From equations (2.32 and 2.33) using equation (2.28) :

e.@-e-=pe - (2.36)

From equations (2.3 1 ) and (2.32):

e. [e ,8+8-1J= [1+e, ] p8 - - (2.37)

From equations (2.30 and 2.33):

b) Case 8 s 1.0 ( w s a,) b

It can be shown that.

From equations (2.39 and 2.40) and using equation (2.28):

e, -e,8=p8 (2.43 - -

From using equations (2.39 and 2.3 1):

e: [e.+1+8] =[l+e.IpB - - ( 2 . 4 4 )

From using equations (2.40 and 2.30):

e - [ e . û + l - û ] =[l-e,]pe (2.45) - A -

Some possible conclusions can be made on the equalities and inequalities

formulated above:

1 ) For the case of 0=1 .O (resonance), the set of equations for 8 r 1 .O and the set of

equations for 0 a 1 .O become identical as expected.

2) The shifl of the fiequency squared to the left ( E, ) is always less than the shift of the

fiequency squared to the right ( E, ). There are two cases when E, does equal E, . One

case is when 8=1.0 (resonance) and E,=E, . The other case is when p=O , for which

E~=€~=O.

The above relationships recornmend that the spread in the square of the coupled

fiequencies is dependent on p . When the modal mass ratio is as small as 0.01, the spread

in the square of the coupled frequencies is more than 20% and the spread in the coupled

fiequencies is more than 10%. Thus a coupled mode1 even in the case (p=0.01) u t M g a

time-history approach wiii produce results which are vey much dependent on the specific

nature of the frequency content of the time-history used in the neighbourhood of the

fiequencies of the system.

Figure 9 shows the relationship of e, (the largest SM in the square of the

frequency) with the mass ratio for different 0 values. The shift in the fiequency is also

shown on a different scale to the lefi. From the figure it can be concluded that the

maximum shift in fiequencies happens at 8=LO (resonance case). For a value of p less

than 1 .O the maximum SM in fiequency is 6 1.8%

~ r o ( ~ r ( m p f o r # I n n n t 0

Fig.9

PROPOSED L'NCOL'PLING CRITERIA

Any uncoupling criteria should iirnit the change of the coupled frequencies from

the uncoupled frequencies as well as the change in response. For a relatively broad-band

spectrum, as used in the seismic analysis of nuclear power plants, lirniting the change in

frequencies automatically limits the change in response generated.

The previously developed equations can be used to develop a new uncoupling

criteria by liniiting E, to be equal to or less than a specified value ofÊ(note that E, is

The uncoupling criteria is given by:

C hoosing E (SM in the square of frequency) as 0.10 and, for simplification, taking (1 -7

as unity;the following new uncoupling cnteria is proposed:

Q- - frequency of secondary system g= ( L) -= ( o. freqüency .of primary system

1:

The above uncoupling cnteria is based on a maximum of 10% shift in the

eigenvaiues (square of the fiequmcies) or alternatively a maximum of 5% shift in the

fiequencies. This Ievel of error is believed to be consistent with other aspects of the

seismic design process. Figure 10 shows the proposed cnteria. In the figure, the dark area

represents the region where a coupled analysis is required, and the white area represents

the region wherea coupled analysis is not required.

ami _ .-• i +

. 4 1

l I

.--o.- m-

I 1 . -. m I m 1

Fig . 1 O-proposed uncouplkg criteria

26

2.3 DISCUSSIONS FOR THE UNRESTRAINED SECONDARY

SYSTEM

There are two dzerent case scenarios for the reaction of the piping system for the

unrestrained 2-DOF system model.

Case A

When the fiequency of the secondary system is higher than the fiequency of the

primary system, the secondary systernxan be considered to be very stiff When the mass of

the primary system moves, the mass of the secondary system moves with it. If the

stifiess of the secondary systern is very hi@, then it can be considered that the primary

system and the secondary syaem are joined together as one. Therefore, only the mass of

the secondary system contnbutes to the motion of the primary system. Hence, the

primary system undergoes a small fiequency shift. This small fiequency shift decreases the

frequency at which the primary system vibrates.

Case B

When the fiequency of the secondary system is less than the fiequency of the

prirnary system , the secondary qaem wiîi not move. Therefore when the mass of the

primary system moves only the stiffness of the secondary system contributes. It shouid

aiso be noted that when the secondary systmi fiequency is less than the fnquency of the

primary system, it means that the secondary systern is flexible and the primary system is

rigid. In this case the response of the secondary system behaves iike a SDOF model. It can

27

also be said that the driving fiequency is the frequency of the primary system, because the

primary system undergoes a large positive frequency SM ( fiequency at which the pkary

system vibrates, increases).

2.4 TWO DEGREE OF FREEDOM SYSTEM WlTH

RESTRAINT

Fiaure A

Fig. 1 l A) coupled qnem B) uncoupled primary system C) uncoupled restrained secondary system

Fiaure C

For the coupled system, the natural fiequencies are given by:

(K is the nifiess, o is the frequency, and M is the mass)

Figure 1 1 (A) shows the coupled system. Figure 1 1(B) and 1 1 (C) show the uncoupled

primary and secondary synems. The secondary systern is attached at the two ends. One

end is comected to the couphg point on the primary system and the oths end to the rigid

support system.

The undamped fiee vibration of this system is represented by the following equation:

The characteristic equation can be written as:

where the coupled fiequencies are represented by o.

The following equations are the transformation equations that conven a rearained

secondary system to an unrestrained secondary ?stem with Equation 2.42 (the above

equation) rewritten as equation 2.43 (the equation below):

1 7 where o,-= (2.44)

- 0'2=012 1 +- ks (2.45)

ml

ms=m2 (- k2 ) (2.46) k2 +k3

4namicJly EquivJmt SDOF nunUy-Unratmincd SDOF Sccondrry Swan

A) Scparated Modified SDOF Runuy Synm 0 ) DyriMlcrlly Equivaicnt Umcsinintd SDOF Sccoaduy Syztcm C) Combineci Systan

Fig 11

Figure 12 diagrarnrnatically illustrates how the coupled fiequencies are cplculated. Fun,

the static renraint due to the secondary syaem,X ,.is rdded to the stifiess of the pnmary

system. This modifies the pnmary system by changing its stifmess fiom k, to k, +K. Second, the secondary synem is rcplaced with an unrestrained secondq syaern having

stiffhess and mass of k, and m,, respeaively. rq is the modal rnass of the secondary system

and is evaluated by giving unit displacement at the support. niird, the coupled hquencies

are calculated using the characteristic quation (2.43-in this chapter). (for simplification,

the original *esses and masses were used in the formula).

Equations 2.44 to 2.49 (in this chapter) represcnt the transfoqation quations

between the two systems. This implies that with the trandomation specified , i11 the

31

design curves developed for Figure 9 can now be used for restrained secondary systems.

In short, when the secondary system has a static restraint, the technique developed

works, but is a two step process. The procedure is to first change the primary system

frequency because of the static restraint and then use equation 2.43 (in this chapter)(note

equation 2.43 in this chapter is identical to equation 2.22 in chapter 2.2).

2.5 DISCUSSION FOR THE RESTRAINED SYSTEM

To find out the fiequency shifts for both cases (case A-when the frequency of the

secondary system is greater than the fiequency of the primary system, and case B-when

the frequency of the prirnary system is greater than the frequency of the secondary

system), the process involves two steps.

First Step

The fiequency of the pnmary systern will increase because if you add a suppon, then it

adds to the stifhess, thereby increasing the freguency of the primary system. The

secondary system is therefore adding regraint to the primary system.

Second Step

The same conclusions apply here as seen in the unrestrained secondary systern models

(case A and case B-as read in chapter 2 section 2.3)

CENERAL CONCLUSIONS FOR CHAPTER 2

1) For the sarne mass m, the effective mass ratio (p) is smaller for the restrained system.

Therefore lower shifis in fiequency wiU occur.

2) If the aifhess k, is rnuch less than k, , then the restrained system is similar to the

unrestrained syaem.

3) If the stifhess k, is much less than k, then the modal mass ratio is much less than 1,

then the difference between the restrained system and unrestrained system is large. This is

because the restrained systern had a small enect, while the unrestrained system had a large

effect .

2.6 DISCUSSION OF RESULTS FOR CHAPTER 2

The analysis indicates that perturbation to the primary system parameters due to

coupling with a restrained or unrestrained system are similar. The restrained secondary

system has the effect of adding stifiess to the prirnary syaem and decreasing the effective

mass ratio. Hence, the differences between results obtained with restrained and

unrestrained syaems become larger

-33-

3.0 THE APPLICATION OF UNCOUPLING CRITERIA TO THE

ACOUSTICS MBRATION PROBLEM

OBJECTIVE

The uncoupling critena that was-developed for the spring-mass system in chapter

2 can be used for acoustic vibration problems, involving pnmary and secondary syaems.

The only difference is that, for the acoustic vibration problem one has to use lumped-mass

approximation, and the tenns for mass and stiffhess are combinations of difFerent

variables. The equivalent mass and stifbess for an acoustic vibration problem will be

shown in this chapter.

Acoustic Resonance Phenomena

By acoustic resonance we mean, resonance of a compressible medium. Resonance

of fluid medium causes small pressure fluctuations about the mean pressure in a fluid

medium. This tem is used throughout this thesis.

Most systems whether they are mechanical systems consisting of a series of

springs, masses or dampers or acoustic syaems consisting of a fluid contained within

bounding pipes have resonance fkequencies. Resonance in a piping system is cornmonly

referred to as "Organ pipe behavïour". AU physical piping systems containing liquid or gas,

have fwidarnental fiequencies and associated mode shapes. These are called "standing

waves". Figure 13 shows examples of standing waves (mode shapes) in a pipe.

PRESSURE STANDING WAVE OF ORGAN PIPE RESONANCE

PIPES O P ~ J Ar BOTH ENDS

OUART€R-WAVf RESON ANCE Q1PES OPEN A t ONE EN0

AND CLOS W AT THE OTHER

-35-

These standing waves cm be excited by a source which is a dynamic pressure or

fluctuating velocity or a combination of both. At resonance it can cause large pressure

amplifications in the piping system and may cause mechanical vibration which can cause

faihre.

When analyshg acoustic systems it is very important to know that the anti-node is

the point of highea pressure fluctuation, and the node is the point of zero pressure

fluctuation. One can convert an anti-node into a node, by placing a stub oc resonator at

that particular point. This concept of anti-nodes and nodes is also true for mechanical or

structural systems, except the variable involved is displacement instead of pressure; in fact

the concept is the sarne. In the case of a vibration absorber, the anti-node would be the

mass on the spring-mass syaem.

Gascous (Steam) System

Acouaic resonance and resulting mechanical vibration of a piping syaem in s t e m

or gaseous system has been previously observed in the industry. This behaviour is caused

by excitation of acoustic standing waves. The source of excitation was a compressor or

reciprocating pump or vortex shedding phenornenon.

Liquid (Water) System

The reported incidences involving this behaviour in water or liquid systems are

fewer compared to the incidences in gaseous systems.

The phenomenon of excessive pressure variations causing unacceptable mechanical

damage to components in fluid conveying piping systems, is not new to the engineering

community involved in the design and operation of such systems.

To satisfactorily address the two issues, one must characterize the pressure

fluctuations believed to be the culprit, mode1 the system experimentally and/or analytically

to ensure that one can fùlly understand the behaviour of the system, and then find a way of

dealing with the pressure fluctuations.

In such a system the response depends on the excitation fiequency and the degree

of damping inherent in the system. If the excitation source is white noise the system will

respond at its natural fiequency or one of its higher harmonies (modes). For the case

where the system is excited continuously at or near its natural fiequencies the response of

the system may be very large and only limited by the damping inherent in the system.

Acoustic Resonance In Piping Systems

It has been observed that resonmce of branch lines in a large piping system is

usually the cause of excessive pressure pulsations or vibration of piping systems. This

-37-

happens whenever the frequency of the branch pipe coincides with the fiequency of the

main piping system. It is therefore, important to develop simplifled analytical method to

deai with this problem.

How much branch pipe resonance cm affect the resonance of the main pipe can be

studied by using the uncoupling criteria descnbed in this thesis. This is done, as in

smictural systems, to assess if uncoupling of branch pipes is acceptable.

The excitation (hamonic and steady-state) for the acouaic vibration problems

faced at Darlington Nuclear Station is caused by a pump. The kind of pump used is a

centrifirgal type pump (fan impellor system), which is classified as p-source (pressure

fluctuation). This pump produces pressure pulsations at the vane passing fiequency

(rotating speed x number of vanes). The other type of pump, which can dso be used, is a

reciprocating pump which is classified as v-source (velocity fluctuation).

3.1 DARLINGTON NUCLEAR GENERATING STATION

ACOUSTIC VIBRATION PROBLEM

DARLINGTON primary heat transport system consias of large primary piphg

systems and a number of smdler piping systems. The large system consias of large pipes

and headers. The smdl system consists of a small diameter pipe and fuel charnel. There

are 120 of such assemblies co~ec ted to one header.

The pump generates pressure pulsations at 1 50 Hz of sigaificant amplitude. This is

caiied the vane passing frequency.

38

blade or vane passing fiequency = (rpm/60) x (number of vanes) = excitation at this

blade passing fiequency

exampie:

Darlington rpm = 180Q Blades=S

there fore,

blade passing fiequency = ( 1800160) x 5 = 1 50 Hz.

It was found that the larger system had an acoustic resonance fiequency of 150

Hz which amplified the pressure pulsations (this was done by ABAQUS modelling and in-

plant testing and measurements) . Thus large pressure pulsations were observed in the

header. Pressure pulsations in the header act as a source for pressure pulsations in the

individual feeders. Depending on the resonance of the feederlchannel assembly, the

pressure pulsations in the channel are further arnplified relative to the header. Large

pressure amplification occurred when the resonant of feeder-channel assembly was located

at the high pressure location of the header profile. In channels where significant

amplification of pressure pulsations occurred, fuel failure occurred.

To analyse this problem, one requires a very large simulation mode1 consisting of

the main piping system and al1 (1 20) feeder-channel assembiies. This becomes quite a large

task. The task can be simplified if the uncoupling criteria, developed in chapter 2 of this

thesis, can be used for analysing this problem. Therefore by using a lurnped-parameter

approximation (shown later in this chapter), the fiequency shift of the primary piping

system can be predicted by using the uncoupling criteria graphs in chapter 2. Hence the

Darlington system cm be analysed separately as primary and secondary syaems.

39

Figure 14 shows a typical feeder and header arrangement at Darlington Nuclear

Station, and Figure 15 shows the 37-element fuel bundle.

u 1. AEACTOA OUTLET HEADER 2 UEACTOA INLR HEADER 3 FEEOER TUBE SPRiNG HANCERS /- 4. CALANDRIA EN0 SHIELD FACE

S. SUPPORT ORACKEtS 6. WALKWAY 7. EN0 FlnlNGS 4 m m GENEAATORS O. INSUUTION CABINET

10. -€AM GENERATOR SUPPORT COLUMN

Figure 14 - Heat Transport System - Typid Feeder rnd Heada Anangement.

42

3.2 EFFECT OF TRANSMISSION LOSSES

For any pipe system there are losses attributed to the:

1. Pipe entrance or exit

2. Sudden expansion or contraction

3. Bends, elbows, tees, and other fittings

4. Valves,open or partially closed

5. Gradua1 expansions or contractions

However, al1 of these losses are a part of the viscous losses (damping) in the system.

3.3 EFFECT OF VISCOUS FLOW IN PIPES

The effect of viscous flow within confining walls such as pipes is discussed in this

section. Viscosity in the acoustic system is quivalent to damping in the mechanical

system. It must be pointed out that no general analysis of fluid motion yet exias; but there

are several known particular solutions, specüic digital cornputer solutions, and lots of

expenmental data. There is continuing research, but it is expected that much t h e will pass

before any general theory is produced on the specific effects of Mscosity on this research

topic.

An interna1 flow is constrained by the bounding walls, and the viscous effects will

grow and meet and permeate the entire flow. Although, this is a fact, it must be stated that

higher damping means a lower response. Therefore the effect of the pressure pulsations

wiil be lower. That means the endplate might not break on the fuel bundle. But since the

43

viscosity can't be increased or decreased in heavy water, this is a variable we cannot

control. me viscosity ofheavy water is simikar to thut of top water)

3.4 EFFECT OF PIPE FLOW

The flow in the feeder channels at Darlington Nuclear Station is turbulent and the

pipes are considered to be smooth. One of the assumptions made when denving the

simplified acoustic equation from the compressible Navier Stokes Equation, is that the

mean velocity is much less than the sonic velocity. Since the flow rate is smailer than sonic

speed, it does not have an affect on the frequeocy and mode shape. However, the flow

rate has effects on the amplitude of the vibration-through the darnping term.

3.5 EFFECTS OF DIAMETER OF PIPE

The diameter of the pipe does not have a significant effect on the frequency and

mode shape. For a uniform tube changing the diameter of the pipe does not change the

frequency because it gets canceled out in the wave equation.

3.6 ATTENUATION (FOR PRESSURE PULSES IN THE SYSTEM)

THE pressure amplitude decreases as the wave propagates. This does not afEect

the uncoupling criteria. In the uncoupling cnteria the amplitude is not considered. The

theoretical attenuation laws state that there is a correlation between what happens at the

44

source and at the site of failure. But generally the intensity at the failure site decreases

with increasing distances between the site and the source.

3.7 REFLECTION (COMPRESSIVE WAVES)

IF the liquid is turning 90 degrees through an elbow, then the wavelength is so

large that there is literally no reflection. If there is a closed boundary condition then the

wave reflects with the same sign. If there is a open boundary condition, then the wave

reflects with the opposite sign.

3.8 AXIAL MODES VS. RADIAL MODES FOR THE FLUID

The analysis in this thesis uses axiai modes. The assumption is made that the

distribution is the same across the cross-section of the pipe. The axial modes are easier to

excite than the radial modes, because the radial frequency modes tend to be very high

fiequencies. An example is to illustrate this is as follows:

I f L = 10 ft,d = lft,

1) ?L=(2 x L)=ZOft axial fiequency = f = (c 1 A)

= (1000/20)

= 50Hz

2) h = ( 2 xd) =2tt radial fiequency = f = (c l ic)

= f =(1000/2) = 500 Hz

45

As you can see, the axial frequency is lower, therefore the axial modes are easier to excite.

46

4.0 CONTINUOUS SYSTEMS

Structural systems (e.g.piping systems) and acoustic systems are continuous

systems. Before we start the uncoupled analysis of acoustic systems, it must be

understood that a lumped-mass system as a mathematical idealization of a continuous

system. The theones developed can be used to get insight into the dynarnic interaction of

continuous systems.

The previous analysis shows the uncoupling criteria for single degree of fieedom

structural systems (lumped parameter systems). A continuous system is a multi degree of

fieedorn system. A continuous system can be replaced with equivalent-lumped parameter

systems to assess the interaction of primary and secondary systems. This study is an

approxhate analysis but does provide a usefùl tool to get insight into the dynamîc

interaction of the two systems.

The uncoupling analysis was done on an acoustic system (continuous system).

The only dEerence is that a continuous system is a multi degree of fkedom system and

requires use of modal aaalysis techniques. These modal analysis techniques are relatively

complex procedures that involve the use of participaction factors etc. This was beyond the

scope of this thesis. However the application of the uncouphg aiteria WU be shown with

ABAQUS Runs (finite element software).

On the foiiowing pages we will show the approximation of continuous systems by

lumped-mass systems.

4.1 MATHEMATICAL IDEALIZATION

A lumped-mass vibration system is a mathematical ideabation which forms a

usefùl approximation to an actual system. An actual system has its mass and its elasticity

distributed throughout; a lumped-mass system has its mass lumped into ngid bodies, and #

its elasticity concentrated into massless springs. In the pst, attempts have been made to

approximate continuous systems by lumped parameters; some of these are discussed in the

following sections.

4.2 APPROXIMATION OF CONTINUOUS SYSTEMS

Most problems concemed with the vibration of continuous systems can oniy be

soived by applying numencd methods to an equivalent system. The success of the analysis

depends on how closely the behaviour of an actual system c m be modelled by an

approximate system [47]. There may be many methods to approximate the continuous

systerns depending on the problem but the few important ones are:

1. Lumped-parameter approximation.

2. Fite difference formulation.

3. Fite element techniques.

In this repoxt we wîii disaiss the lumped-parameter approximation, and the frequency shift

results wiU be verified using ABAQUS (finite elernent program).

48

4.3 LUMPED-PARAMETER SYSTEMS

Bishop, GladweU and Michaelson [32] have given a good account on the

approximation of continuous systems by lumped-parameter syaems. There are various

modek which can be used for approximating the continuous syaems by lumped

parameters. Duncan [48] has shown that if the mass of an elastic body of linear fonn is

condensed into rigid masses or particles situated at the midpohts ofequal segments,

while the elastic properties of the body are unaltered the error in the fiequency of any

natural mode of oscillation varies inversely as the square of the number of segments. When

the masses are not placed at the midpoints of the segments the error varies inversely as

the number of segments. The errors in the natural fiequencies for the approximation aIso

Vary inversely as the square of number of segments.

In a lumped parameter system, the continuous system is replaced by an equivalent

system consisting of concentrated masses and stifhesses. The lumped rnass and the

&ess are estimateci such that the Kinetic Energy and Potential Energy of the two

systems are equivalent.

49

4.4 DEMONSTRATION OF SIMILARITY BETWEEN LUMPED

PARAMETER SYSTEMS AND CONTINUOUS SYSTEMS

The foiiowing derivation shows that the lumped-parameter approximation is good

for continuous systems for the first mode of vibration. This following proof is valid

because it was proven that the stifniess for a stnictural continuous system ( a bar ) is K=

( A m ) . In the proof, the lumped mass case is on the left hand side of the equation, and

the continuous system is on the right hand side of the equation.

definitions:

dm= dx A (unifonnly distributed mas)

To get lumped stifhess

SThUN ENERGY ANALYSIS defenition : L = strain energy

1 L~ U 2 KU^ = I-(-) Adx 2 0 2

The general lumped parameters can be expressed as:

M-apAL (where M is the mass)

K= (PAE)/L (where K is the stifhess)

where a and p are constants and Vary based on the displaced s

For the linear displaced shape that we have used the constants are-

a=lB and P=i

for u = ( Ü ~ ) x

4.5 FINITE DIFFERENCE FORMULATION

The one dimensional equation goveming the srnail free acoustic vibration p(x,t) of

a pipe (which is liquid filled) is:

where p is the density of the material. k is the buik modulus, A is the cross-sectionai area.

For vibration at frequency a, we assume

p(x, t) = P ( x ) exp ( i o t )

Substituting (4.5 1 ) into (4.52), we obtain

Writing Eq.(4.54) for a particular point i, we get

d dP d ' ~ [-(kA) ] ; [ - I L + (kA) ; [-1 , = - A , ~ , G J ' P , (4.55) dx dx - dx- *

Now, use the foilowing finite difference approximations for the denvatives in Eq. (4.55)

where

Cornparhg Eq44.56) with the generai equation of motion

This can be written as

[k-hm] p=O

where ~=o'

This means that the approximation of continuous systems by finite Merences leads us to

the problem of lumped-parameter systems. Therefore, lumped parameter approximation is

a very good mathematical idealization of a continuous system.

5s

4.6 ANALOGY BETWEEN ACOUSTIC AMI STRUCTURAL

SYSTEMS

The differential equations used to represent an elastic vibration of a rod and

acoustic vibration of a fluid are of the same form. Therefore, the uncoupling criteria

developed for structural systems is also applicable for acoustic systems. The following

section will give the derivation of the partial differential equation for a structural system (a

bar), and show the similarity between the wave equation and the differential equation for

vibration of a bar.

4.61 Longitudinal Vibration of Rods

The rod considered in this section is assumed to be thin and uniform dong its

length. Due to axial forces there will be displacements u dong the rod which will be a

function of both the position x and the tirne t. Shce the rod has an infinite number of

natural modes of vibration, the distribution of the displacement will differ with each mode.

La us consider an element of this rod of length dx (Figure 16).

Figure 16 - Displacement of nid el-

57

Lfu is the displacement at x, the displacement at x+dx will be

It is evident then the element dx in the new position has changed in length by an amount

and thus the unit strain is

Since fiom Hooke's law the ratio of unit stress to unit strain is equal to the modulus of

elasticity E, we can write

d u - P - (4.61) zx AE

where A is the cross-sectional area of the rod. DEerentiating with respect to x.

We now apply Newton's law of motion for the element and equate the unbalanced force to

the product of the mass and acceleration of the element

where p is the density of the rod, mass per unit volume. Eliminating

a P / &

between Eqs.4.62 and 4.63, we obtain the partial differential equation

d'u - -- d2 u ( E h ) - a t 2 ax2

where

acoustic systern

(wave equation)

d2p _ 1 d'p ---- d x 2 c2 d t 2

- - - -

buik modulus, k -- --

young's modulus, E

As can be observed the partial differential equation for a rod or bar (a continuous

syaem) resembles the partial differential equation for an acoustic system (ah a

continuous system). The only dserence between the bar and the acoustic system is that

for the bar, one-deais with displacements, and for the acoustic systern, one deals with

pressures. One minor dSerence between the two systems is that with the bar, one uses

Young's modulus of elasticity, E, and with the acoustic system, one uses the Bulk

Modulus, K.

4.7 EQUIVALENT MASS AND STIFFNESS PARAMETERS

When one does uncoupling analysis of a continuous system it is important to

understand what are the equivalent rnass and m e s s parameters for it. A bar is a

continuous system tike an acoustic systern. The equivalent mass and stifiess for a bar and

acoustic system are given below in the table. The confimms acou~lic system has to be

converted to a lumped patmeter system for doing any uncoupltg m i y s i s .

Table 1

This table is important because the uncoupiing criteria that was done for the

lumped-mass system applies tu the acoustic system with the equivalent rnass and f i e s

paramet ers.

The parameters of the acoustic system were determined by inference. The accuracy

of the shifts predicted by the uncoupling criteria depends on the assumed displacement

pronies.

mass

stifiess

Acoustic Sysîem

I

(w k)

(A PL) .

Spring-mas System

m

k

Bar

Plu

( A W

61

The previous table was based on the fàct that analogies have always existeci

between dEerent fields of engineering study. The analogy that was used in this study was

show so that the uncoupüng cnteria could be used to help the engineers solve the

acoustic vibration problems at the Darlington Nuclear Power Station.

62

4.10 UNCOUPLING ANALYSIS OF ACOUSTIC SYSTEMS BY

ABAQUS RUNS

A finite elernent program called ABAQUS (a very popular commercial program)

was used in this section so that the author could ver* that the lumped parameters

developed for the acoustic vibration problem were correct. The results of the output

verifed that the lumped parameters were correct, because the example models (chapter 4,

section 4.1 1) behaved as predicted (in terms of the fieguency shifts of the pnmary

system). Hence, the two graphs that were developed in chapter 2 are applicable for

acoustic vibration problems (including the Darlington Station fiiel failure problem).

Description of ABAQUS

ABAQUS is a cornmercially available general purpose finite element analysis

cornputer code, which can be used to perform a wide variety of advanced structure and heat

d e r a n . . Hibbitt, Karlsson, and Sorensen, Inc .[13] , is the dweloper of ABAQUS.

ABAQUS is used to solve Multidegree of fieedom problems (continuous system problems).

Wtth respect to acoustic modehg and anafysis, ABAQUS provides a set of 1/2/3-

dimensional elements for modehg an acoustic medium undergohg d pressure variations,

and also includes Undace elemmts to couple these acoustic elements to a structural model.

63

These elements are provided primarily so that steady-state h o n i c (ha) response

analysis (passive fiequency response anaiysis) can be performed for a stmctural-acoustic

system, such as in the study of the noise level in a vehicle. The acoustic medium may be

flowing within a rnatrk material that offers resistance to the flow, thus providing some

dissipation. In addition, rather general boundary conditions are provided for the acoustic

medium, including absorbing boundaries. The flow resistance offered by the rnatrix matenal

and the propaties of the absorbing boundaries may be tiinctions of frequency in steady-state

response analysis, but are assumed to be constant in the direct integration procedure [13]..

The the userdefined parameters d e s c n i in the formulation documentation that are

used to specify the properties of the acoustic medium and boundary conditions in an

ABAQUS acoustic analysis are highlighted below.

in an ABAQUS acoustic analysis, the user is dowed to specw

the density of the acoustic medium or fluid, and

the bulk moduius of the nuid. The bulk modulus is related to the density of the fluid and c,

the speed of sound in that fluid by:

64

In addition, the user is aiiowed to introduce acoustical damping by speçifyng r, the

'~olumetric drag coefficient", which is defined as the force per unit volume required to

drive the fluid through a resisting medium at unit fiuid velocity.

W1th respect to acoustical boundary conditions, the default boundary condition is

capped or closed end (Le., zero flow) condition. A non-zero flow or velocity boundary is

specified by coupling to structure through the use of interface elements7 then specifying

the velocity via the structural degres of fieedom. To speafy an open end, the acoustic

pressure is set to zero. A pressure boundary condition is s p d e d by setting the acoustic

pressure to the expected pressure.

Training

ABAQUS is a relatively difiicult software program to use that takes time to use

properly. The input deck has to be properly structured in order to achieve good results.

(ABAQUS uses an intemal routine whkh is based on a variational approach to solve

problems.)

65

4.11 DEMONSTRATION OF APPLICABILITY OF

UNCOUPLING CRITEFUA TO ACOUSTIC PROBLEMS

In this chapter, acoustic qstems were uncoupled.. The analysis proved that the

results obtained for the uncouphg criteria developed for structural systems is applicable

to the acoustic systems (continuous systems). Appendix B contains the output decks

which is an echo of the input decks

Two examples were analyseci to show applicabüity of the uncouplhg critena

developed to acoustic problems. In the firn problem, the unrestrained secondary system

was used, while in the second problem, the secondary systern was restrained. The

significance of these exarnples will be explained in the discussion of results section (4.14).

The shifls predicted by the formulas (for Figure 9) were benchmarked using

ABAQUS, a generai purpose finite element program. Exampie 1 gives the specifications

of the problem for the unrestrained acoustic system. (The specifications were input into

ABAQUS, and ABAQUS computed the uncoupled and the coupled fkequencies so that

the frequency shift of the prllnary system could be calculateci).

Eumple 1-SUMMARIZATION OF ABAQUS COMPUTER INPUT AM)

OUTPUT

For an unrestrained amustic system Uncoupled Runary System

A, = l,Lp = 1 0 , ~ ~ = 1000,Kp = 1.OE9

Uncoupled Seccmdary Systern

A,=O.l ,L,=10.0,p,=900,K,=l.OE9 From the a h input the following fiequaicies were

derived O, =25 Hz o s = 26.352 Hz The above infirmation was used as input into ABAQUS and the following fiequencies were derived: Coupled h a r y System Frequency = 20.506 Hz. Coupled Secondary Syaem Frequency = 30.8 Hz Therefore the fiquency shifis are

Therefore, the primary system fiequency went down while the secondary system fiequency went up .

Figure 17 is an üiustration of the unrestrained acoustic system.

- closed, P m

Secondq Synem

.

I c losed, P m a x

Figure 1 7 - Acoustic mestrained man

-

The results are given below:

Table 2

To produce this table the uncwpling criteria that was developed in chapter 2 was

mass ratio

O. 1

used to predict the frequency shift of the primary systern. It successfully predicted the

fiequency shifl, but with some error. Therefore, the applicabiiity of the uncoupüng critena

to the acoustic vibration probiem was feasible.

ffequency ratio

1.054 1

In these calculations for mass ratios, the a factor was not important because both

primary and secondary systems had the same mode shape. Therefore. this factor wiii

ABAQUS

0.82

cancel out. The calculated shift of the primary system is in the nght direction. It however,

has 25% emor in estimating the shifî. This error is because in making the idealization fiom

Equation 5.24

0.87

the continuous system to the lumped parameter systems, we are missing the mode shape

% error

25%

correction factor. This factor, nonnally, rnakes the shifts predicted by the formulas

conservative for fkquencies other than the first natural fiequency.

To study the lumped parameters derived for acoustic systems, a sensitivity study

was done. The dimensions of the xcondary system were changed as given in table 3.

Table 3

As the table indicates, these changes in dimensions keep the lumped parameters and the

separated secondary system fnquency the sarne. As expected, coupled analysis with each

of these systems gave identical coupled fiequemies. This demonstrated that the lumped

parameters developed were suitable.

In the following example, the secondary syaem was changeci. The Eee end was

now restrained by having a reservoir at that end. To keep the fiequency and the volume of

the secondary system the sarne as before, the lmgth of the secondary system was doubled,

and the cross-section halved. Example 2 (on-the following page) gives the specifications of

the problem for the restrained acoustic syaan.

In this problem, the displaced shape of the primary and secondary system are not

the same. Therefore, the a and f3 factors wili have to be detennined. This was

considered out of the scope of this thesis.

ABAQUS - SUMMARlZATION OF ABAQUS COMPUTER INPUT AND

OUTPUT

For a restramed acoustic system

Primary System A , = l , L p = 10,pp = 1000 ,K, = 1.OE9

Secondary System A , = O.l ,L , = 10.0,pS = 900,K, = 1.OE9 From the above input die following &quaicies were derived uncoupled O ,, = 25 Hz

uncoupleci O , = 26 -352 Hz The above i n f i d o n was used as input imo ABAQUS and the tollowhg fiquencies were derived coupleù primary sysrem fiequency = 23.014 Hz. coupied secaiâary system îrequency = 28.368 Hz Therefore the frequency shifb were

28,368 - 25 sec m&ry system fieirluency shift =

25 = 13%

Figure 18 is an illustration of the restrained acoustic problem.

Reservoir

c losed, P m a x

Rcservoir

Figure 18 - Acounic rcstrained system

4.14 DISCUSSION OF RESULTS OF ABAQUS EXAMPLES

It is usefbl to observe the difference in shif€s between the two systerns. In the

restrained system the shifts were 8% and 13%. This is Iower than the shifis obtained ushg

the unrestrained system. Both shifis were lower because the modal mass ratios were

smaller. Funher, the restraint of the secondary system must have increased the natural

fiequency of the primary system.

As the direction of the shifi has not changed significantly, the effect of restraint

must be small.

As you can observe, the unrestrained and restrained secondary systems for the

spring mas system mode1 and the acoustic vibration model, behaved in the same way. The

frequency shifts for the restrained secondary system were lower for both systems.

The frequency SM of the prirnary system for an acoustic vibration example was

calculated using the uncouplhg criteria in chapter 2. It successfilly predicted the shifi,

demonstrating the applicability of the criteria to acoustic vibration problems.

5.0 CONCLUSIONS

The goal of this research study was to provide the uncoupling criteria to solve

an acouaic vibration problem. The research study provided a rnethod, so that engineers

involved with this problem could fhd out whaher to analyse the primary and secondary

system together or separately. Therefore ushg the uncouplhg cntda provided,

engineers might be able to analyse complex systmis separately, minimiPng effort, and

saving money. The objective of this study was never to provide guidance on acouaic

anaiysis, compute the sustainable pressures that the pump should provide, or give any

response calculations. That is why parameters such as losses due to elbows and t-

junctions, were not considered. The question that was answered was "Cm 1 use the

uncou pling criteria to anaiyse prirnary and secondary systerns separately?"

CONTRIBUTIONS TO RESEARCH

There were rnany unique accomplishmmts made by this research study:

1 ) Development of a technique to transfomi a restrained secondary system to an

equivalent unrestrained secondary system. This dlowed use of existing equations for

unrestrained systerns to enimate the coupled fiequencies.

2) IUustrated the technique pictorially for a better understanding of the computatioas (with

examples).

3) Devtloped a link between continuous systems and lumpad-parameter systems and

74

extmded the technique devdoped to d y z e continuous systems through lumped-

parameter systems. This was then used to get insight into the dynamic interaction of

primary and secondary systems. Not only were lumped parameten developed for the

acoustic system, but the applicability of the formulations were clearly demonstrated.

(ABAQUS computationd work)

The uncoupling analysis that was done for the acoustic systems was the sarne as

the structural systems, the only difference being the texms used for the analysis.

4) Sirnilarity was shown between structural and acoustic systems and parameters were

developed to use in these equations.

A.H. Hadjian AND B.Ellison, Decoupiing of Secondary Systems for Seismk Analysis, Transactions cf the ASME, Vol. 108, February 1986, pp.78-85.

A. H. Hadjian, Seismic Response of Structures by Rersponse Spectnim Method,Nuclear Engineering an De~ign,Vol.66~No.Z,August 198 1, pp. 1 79-20 1.

A. J. SaIrnonte, Consideration on the Residual Contribution in Modal Analysis, Earthquake Engineering and Structural Dynamics, Vol. 10p 1982,pp.295-304.

A.K. Gupta and D.C Chen, A Simple Method of Combining Modal Responses, Transactions, Seventh International Conference on Stnistural Mechanics in Reactor Technology, Paper No.K3/lO, Chicago, August 1 983.

A.K. Gupta and J.M. Tembulkar, Dynamic Decoupling of Secondary Syaems, Nuclear Engineering and Design, Vol. 8 1, September l984,pp. 3 59-3 73

A.K Gupta, seismic Response of Multiply Co~ected MDOF Primary and Secondary Systems, Nuclear Engineering and Design, Vo1.81, September 1984, pp.385-394.

A. K. Gupta, Response Spectrum Method, Blackwell ScientSc Pubiications, 1990.

A.K. Gupta and M.P. Singh, Design of Column Sections Subjected to Three Components of Eanhquake, Nuclear Engineering and Design, Vo1.4 1, 1977,pp. 12% 133

A. K. Chopra, Dynarnics of Structures - A Primer, Earthquake Engineering Research Inaitute, Berkley, California, 198 1.

A.K.Gupta, Seismic Response of Mdtiply Comected MDOF Primary and Secondary S yaems, Nuclear Engineering and Design, Vol. 8 1, September 1984, pp.3 85-394.

Anton, Howard,'Elementary Linear Algebra' John Wiky & Sons, New York, 1987.

Crandail, S .Hm, Mark, W.D . ,'Randorn Vibration in Mechanical Systems' (AcadeMc Press, 1963).

Hibbitt, Karlsson and Sorenson, Inc.'ABAQUS/Standard User's Manual, Pawtucket, RI 02860447, USA 1995

D.W. Lindley and I.R Yow, Modal Response Summation for Seismic 'Qualification, Second ASCE Specialty Confamce on Civil Engineering and Nuclear Power, Vol. VI, Paper 8-2, KnoxMlle,TN, September 1980.

F.E. Elghadamsi and B. Mohraz, Site-Dependent Inelastic Earthquake Spectra, Technical Report, CM and Medwical Engineering Department, Southern Mahodist University, Dallas, Texas, June 1983.

G.H. Poweil, Missing Mass Correction in Modal Analysis of Piping System, Transactions, Fi ih International Coderence on Stmctural Mechanics in Reactor Technology, Paper No. K1 O/3, 1979

G.V. Berg, Seismic Design Codes and Procedures, Earihquake Engineering Research Inaitute, Berkeley, CaMomia, 1983.

G.W. Housner and P.C. Jennings, Earthquake Design Critena (Engineering Monograph on Earthquake Criteria, Stmctural Design, and Strong Motion Records, M.S. Agbabian, Coordinating Editor,), Earthquake Engineering Research Institute, Berkeley, California, 1982.

Hagar,W. W.' Applied Numerical Linear Algebra', Prentice Hall, New Jersey, 1988.

J. L. Buchanan and P .R. Turner, 'Numerical Methods and Andysis', McGraw-Hill, New York, 1992.

I .L Sackman, A. Der Kiureghian and B.Nour-Omid, Dynamic Analysis of Light Equipment in Structures: Modal Properties of the Combined Syaem, Journal of Engineering Mechanics, ASCE, Vol. 109, N o 1, Febniary 1983, pp.73-89.

J.hl Nau and W.J. Hall, An evaluation of Scaling Methods for Earthquake Response Spectra, Stmctural Series, No.499, University of Illinois at Urbana-Champaign, May 1982.

M.A. Biot, A Mechanicd halyzer for the Prediction of Earihquake Stresses, Nulletin of the Seismological Society of America, Vo1.3 1, 194 1 ,pp. 1 5 1 - 1 7 1

N.M.Newmark, J.A.Blume and K.K Kapur, Seismic Design Criteria for Nuclear Power Plants, Journal of the Power Division, ASCE, VoI.99,1973,pp.287-303.

Pickel, T. W., Jr. 'Evaluation of Nuclear Systems Requirements for Accommodating Seismic Effects' . Nuclear Engineering and Design, Vol. 20, No.2, 1972.

R.P. Kennedy, Recommendations for Changes and Additions to Standard Review Plans and Regulatory Guides Dealhg with Seismic Design Requirements for Strctures, Report prepared for Lawrence Livermore Laboratory, Published in NUREWCR- 1161, June 1979.

R.Riddeli and N.M. Newmark, Statistical Analysis of the Response of Nonluiear Systems Subjected to Earthquakes, Structural Research Series, No.468, Department of

C i d Engineering, University of IUinois at Urbaaa Champa&, Uhana, IUinois, August 1974.

R. W. Clough and J. Pemien, Dynamics of Structures, McGraw Hill, New York, 1975.

S.H. Hayashi, H. Tmchida and E. Kurata, Average Response Spectra for Various Subsoil Conditions, Third Joint Meeting* U.S. Japan Panel on Wind and Seismic Effects, U m Tokyo, May 197 1.

Singiresu, S.Rao, 'Mechanical Vibrations (2nd Edition)', Addison-WESLEY Publishing Company, Don Mills, Ontario. 1990

United States Atornic Energy Commission, Design Response Spectra for Seismic Design of Nuclear Power Plants, Regulatory Guide, No. 1.60, 1973.

R.E.D. Bishop, G.M.L. Gladwell and S. Michalson 1965. The Matrix Analysis of Vibration. London: Cambridge University Press.

Ontario Hydro Repon, Fuel Failure Problem, Regulatory Guide, No.2, 1995.

Ontario Hydro Repon, Support Information Guide, No.6, 1993.

Ontario Hydro Report, Technical Pump Guide, No.45, 1994.

Ontario Hydro, Standard Guide for Huid Mechanics, No. 1, 1990.

D.K.Vijay,'Some Inverse Problems in Mechanics', University of Waterloo, 1972

W.I. Duncan 1952 Q.J. Mech. And Appl Math. 5,97. A criticai examination of the representation of massive and elastic bodies by systems of rigid masses elastically co~ected. .

APPENDIX A

On the fo1lov:ing pages are sorne illustrations of the primary piping systems and the secondq piping syaems. These pictures of the piping systems were taken at Darlington Nuclear Station. There are ais0 some generic pictures of some nuclear applications that help the reader understand the bigger picture of this research.

Picrures of the primary piping systems and the secondary piping systems.

APPENDIX B

The computer input and output represmts the acoustic vibration runs that were made to find out the frequency shift of the pnmary system due to the secondary system.

V U J

* EEAD ING COWLEE SYSTEM, NATURAL FREQUENCY * * NEUTRAL FILE GENERATED ON: 19-FEB-92 14:18:14 PATABA VERSION: 3.1A * * * * t

NODE DEFINITIONS

"SOLID SECTION, ELSET=PRI, MATERIALfPRI I.G * S @ L I D SECTICN, ELSETnSEC, MATERIAL-SEC C.35 * * nY!TERIAL, NAME=PRI *SZ!ZSITY 1 C O O . 3

"ACO'JSYIC MEDIUM,BULK MODULUS l.OZ9, R R

*LYr,TERf AL, NAME-SEC *DEXSITY

900. O *ACOUSTIC MEDIUM,BULX NODULUS 1. CE9, n u

*S'TEP *FREûwENCY 5 , 4 0 0 . 0 , " SOUNDARY 1, 8 , 8 , 0 . 0 2 0 1 , 8 , 8 , 0 . 0 *NODE PRINT, FREQUENCY=l POR, ??OR *EL PRINT, FREQUENCY=O WODE FILE, FREQUENCY=l PC3, PPOR *END STE? * n

Page 1

*HEAD ING COUPLED SYSTEM, NATURAL FREQWENCY * * NEUTFAL FILE GENERATE0 ON: 19-FE9-92 14:18:14 PATABA VERSION: 3 .1A * u * * NCDE DEFINITIONS * * *NGDS 1, c, c 1 C l , i 0 , O 201,50,0 'ElûZN 1,101 101,2G1 "ELEMENT, TYPE-AClD2 1, L 2 * ELGEN 1,200 * ELSET, LLSET=PRI, GENERATE 1,100 *ELSET, ELSET=SEC, GENEPATE ICI, 2 3 0 * t 'SOLI3 SECTION, ELSET-PR1 , MATERIAL=?Rf 1.9 * SCLID SZCTION, ELSETxSEC, MATERIALcSEC 3.923, * n

*.XAT33IA3, NAME=PRI *DZNS ITY 1000 .O RACCUSTIC MEDIUM, BULK XODULUS l.OE9, t u

"YA'XRIAL, NAE?=SEC *f ENS IX

2 î S . , *ACOUSTIC MEDIUM,SULX MGDULUS î.OE9, * n

t t

R X

* * **----,---------------------*----- ZC * u r

* S E P * FREQUENCY 5, 400. O, *SOUNDARY 1, 8,8,0.0 201,8,8,0.0 *MODE PRINT, FREQtfENCY=l PGR, OPOR *EL 2 X N T , FREQUENCY=O *NODE FILE, FREQUENCY=l ?OR, PPOR "END STE? u *

- $6SDRB2:[ARVIND.ACOUSTIC.BENCH.RC\J.CONS]PRI.NP;l 4-FEB-1997 08:03 Page 1

*HEAD I N G PRIMARY SYSTEM UNCOUPLE FREQUENCY * * NEUTRAL FILE GENERATED ON: 19-FEB-92 14:18:14 PATABA VERSION: 3.1A * * * * NODE DEFINITIONS * * "NOCE r,o,n 101,10, O *NGEY i, 101 *ELEY?NT, TY?E=AClD2 L 1 , 2 "ELGEN, ELSET=PRf 1,100 R * *SOLID SECTION, ELSET-PRI, MATERIALsD20 1.0 * * "MATERIAL, NAME=D20 *DENS ITY 1000.0

*ACOUSTIC MEDIUM,BULK MODULUS 1.0F9,

X R

* f " S TEP * FEGL'ENCY 1,40C.O, *SOL.DARY il 9 1 8 p 1 . 0 * N O E FRINT, FREgCENCY=l ?C)R,??GII *EL ZRfNT, FFECUENCY=O *NODE FILE, ErlEgWENCY==I FOR, PPOR "END S'TE?

* HEAD ING SECONDARY SYSTEM UNCOüPLE FREQUENCY * * NECTRAL FILE GENERATED ON: 19-FEB-92 14:18:14 PATABA VERSION: 3 .1A * * '* NODE DEFINITIONS 1) * "NODE 1,0,0 101,25, O 'NGEN ï, 101 *ESLLCEXT, TYIE=AClD2 1,1,2 *ELGEN,ELSET-SEC 1,190 f *

'SOLID S X T f O N , ELSET=SEC, MATERIAL=DZO 0.05 * * *F!TE3IAL, NAME=G20 'CENS ITY

9 C C . O 'ACCUSTI 2 a D L U M , BuLi MODULUS I .2E9,

* * t * * S'TE? *--- r .cQCE?fC? 1,430.C, " BOtTNDMY Ir 8 , 8 , @ . 0 191,8,8,0.0 rNCGF ?RINT, FREQUENCY=l ?CR, F2GR * 5~ PXINT, FEQUENCY=O 'FIODE FILE, FEQüENCY=l PORI FIOR 'ENS STEP

U V U

Page 1

"HEADING SECONDARY SYSTEM UNCOUEJLE FREQUENCY ** NEUTRAL FILE GENERATED ON: 19-FEB-92 14:18:14 PATABA VERSION: 3 . 1 A * R ** NODE DEFINITIONS * * *NODE 1, o r 0 i01,40,0 *NGEN 1, i01 * U T - L-LMENT, TYPE=ACIDZ 1,1,2 *ELGEM, ELSET=PRI l, i00 R * * S O L I D SECTION, ELSET=PRI, MATERIAL=DSO O. 025, **O .O9091 R t

*.hWTERIAL, NAME=D20 *DENSITY

Z Z S . , "ACOCSTIC MECIUM,3ULK MODULUS 1. CP9,

R t

* * "STE? *FXEQüENCY 1, 4OO.G, * a o m m y 1, 8,8,tr.O I G l , 8 , 8 , 0 . @ *NODE PRfKT, FFEQUENCY=: POX, ZPOR *EL PRZNT, FRZQUENCY=J RNODZ FILE, FfiEQUENCY=l POR, ?FOR *END STEP

- $6$DRB2:[ARVIND.ACOUSTZC.EENCH.RAJ.CONS]SEC3.INP:2 Page 1

* HEADING SECONDARY SYSTEM UNCOUPLE FREQUENCY * * NEVTRAL FILE GENERATED ON: 19-FEB-92 14:18:14 PATABA VERSION: 3.LA u *

"SOLID SECTION, ELSETaPRI? MATSRIALaDZC O. 05 u * *.XATEF.IAL, N M = D 2 0 "DENS I T Y 8i8.18 *ACOUSTIC WDIUM,BULK MODULUS 1. E 9 ,

u n

* t *STEP ' S E Q V E N C Y 1,4VC.O, * B O m ? a Y l f 8f8,1.0 :01,8,8,G.3 'NODE PRIX', FREQUENCY=l POR, PPOR "EL P X N T , FflEQLFNCY=O *NODE FILE, F-XQrJEKCY-1 FOR, PPOR "EX3 STE?

sSSsSsss S S s ssssssss

9 S S

ssssssss

OH Imam OS-00-2D-U-IO-U, YOU IW AVMORfZm TO RERl ULDLICWDYü OOSt D E . 31, 1997

ms, XNC* 1080 Iurn Str-t Dartnckmt, RI 02060-4847 TaA: 101 727 4200 tu: 401 727 4200 mil : supportehkr .cm

iiu (WEST), niC. U S ( K U T G W ) , flc.

::::.";o5" L W l l b r d 3 0 2 0 1 O , ~ k h R 0 8 d l o t t a 210

Ilrruk, U 94560-5241 T.aington U l l r , iR T ~ J : 510 794 ssst dam-2218 tu: 510 794 US4 ?d: O10 932 4202 *il: hlmrtehkr.cair tu: 110 932 8204

-1: weui.c~

. * * t * * * * * t + 8 8 . . * t * t * * ~ * a * e a * . *

..+.te*..* t .0 *m.. r N O T I C E . .*+*****.**.a**.. t . t .

n r s 1s aMQus vrruraat 5.5-1 t

8

O L U 9 t l Q U O ~ m U ~ O f I I C W U I O l S . S l Q U O I L t S l PLUS T U HûTES AC-MïIlG T U S W. TI= Bû2XS l

* CA# U Ob- DY OSXZfG TB1 flOFOUQLnOW ORTfûli OQ T U * ~ U S ~ L X l e t .

SEC

PPPPPPP 0000000 0000000 0000000 O O O O O O O 0000000

QF9QQ95 0 0 0 0 0 0 0 0000000

oooop * . * . O O O O O 00000 0 0 0 0 0 00000 00000

m n m n n Ô 9 & & 0 ooooo

T O I M A T O R A L ? R t Q U t E C X t S

rmDt NO

1 2 3 4 5

nont NO

1 2 3 4 5

rmOt iso

1 2 3 4 5

TOTAL

PDOR

0.7606 0.7712 0.79Sl 0.8115 0.8272 0.1423 O . 1568 O . a706 0.8839 O . 9963 O . POO1 O . Pl92 O . 9297 o. 9395 0.9405 O , 9568 0.9645 O. 9713 0.9775 O . 9929 0.9876 0.9915 O , 9947 O. 9972 0.9989 O . 9998

1.000 O . 9994 0 . 9 9 0 0.9960 0.9932 O . 9997 O . 9854 0.9803 O . 974s 0.9690 O. 9607 O. 9520 O * 9441 0.9347 0.9246 O . 9x30 0.9023 O . 8901 0,1772 0.0637 0.8496 O. 8349 0.8194 0.8034 O. 7967 0.7695 O. 7517 0.7331 0.7144 O. 6949 O. 6749 O. 6 4 4 0.6334 o. 6119 o. 5900 O . 5676 O . 5448 o. 5216 O. 4900 O.47U 0.4497 0.4230 O. 4001 0 .31 48 0,3492 0,3233 0.2013 0.2710 0.2444 O. 2 l77 O . 1909 0.1639 O. l368 O . 1095 a. 222lt-a2 5.48491-02 2.7435t-O2 7.6207t-42

MODE TOOT- BOR DROR Non

1 N TOLLOWING tllfL Il PI1IRID TOR A U UORW

=DE ?OUF- BOR PPOR uaTt

1 HODI POOT- RQR PPOR N o m

APPENDIX C

DESCRIPTION OF APPENDIX

On the following pages are al1 the derivations for the fkequency shift equations for the combined and separated syaems.

Drvcloping Frequency Equation For Piping Systems

(Supporicd SYstem)

Devdoping the gc iwi l fn~uency equation for the system

Sirnpfifying furtbur:

Setting the vaiue of mUml to a single variable: (p)

nie rsuitnnt equation is-

Devdoprnent Of Equation For I,

Developing the quation for f, so it cia k uscd to plot the

DERWAT~ON OF EQUATIONS FOR FREQUENCY S m S

Derivation o f second frquenq s u t equation

Simplifying

The r d t a n t equation is

The r d w t equation is-

Simplifying (Definition of m on p d o u s derivation)

The rtsuitant equation is-

APPENDIX D

DESCRIPTION OF APPENDM

On the foilowing pages is a iterative computer program that iterates the m a s ratio and the frequency ratio of the ~Mnary system and secondary systern und the 10% shift in fiequency critena is met. The results of this program were used to draw the graph (Fig. 10) on page 25.

PROGRAM TO CALCULATE FREQüENCY SHIFï GRAPHS [S FiGURES 3 AND 5.

THE FOLI.OWNG PROGRAN IS AN ITERATIVE PROGRAM THAT ITERATES TI= MASS RATTO A\D THE FREQUENCY RATIO OF THE PRiMARY SYSTEM .OR3 SECONDARY SYSTEM WiL THE IO% S?üFl' iN FREQUENCY CRITERIA IS MET.

RM IS THE MASS RATIO RF IS THE FREQUENCY RATIO RR 1 S ( 1 -FREQUENCY SlUFT) RR SHOUtD BE 0.9 TO GRAPH IT

REAi A,BRF.RRF22JULS.T,RM OPEN (UNIT= I O)JILE= 'WCTORY.FOR'.STATUS='WORY') RF*. 1 RM4.3 A= 1 .O

9 ffZ=RF"RF B= 1 +RF2+RMmW2 R=((B~0824*AmRF 2)'*O.J)/PA)**O.J S= l sRMI( I -R8*URF2)**2 T= 1 +MW( 1 -R0*2/RF2) RR=T/S &RITE ( 10.50) R M N R R R

50 FORhMT(F9 4,3X.F8.4.3XE9.6.3X.F9.6)

ELSHF ((RR, LT.0 91) THEN GOTO 1 1

ELSF, RV=RM+ 1 .O GOf O 9

ESDiF 11 RF=RF4.2

R\f* 1

13 CLOSE (UPu?T= IO)

STOP mi

TEST TARGET (QA-3)