Preamble

Design ethodology traight Fluid-Conveying uclear tochastic ibration Tian-lin Yong-fu Liang Liang

1 College

of Nuclear Science and Technology, University of South China, Hengyang 421001

2 College

of Mechanical Engineering, University of South China, Hengyang 421001

3 Key

Laboratory Advanced Nuclear Energy Design Safety, Ministry Education, Hengyang, 421001, China author. addresses:

Abstract

tochastic vibration nuclear power plants amplify cyclic stress, vibratory displacement velocity response straight fluid-conveying pipes primary address this, stochastic-vibration mathematical model clamped clamped straight pipes under stochastic excitation established model integrat stochastic vibration theory finite-element method.

Consequently, systematically quantif pipe-support locations structural parameters influence natural frequencies stochastic displacement, velocity stress responses fluid-conveying straight pipes.

Relocating support along first increases decreases natural frequency, reduce natural frequency stress enlarge displacement velocity amplitudes. larger internal diameter suppresses displacement, velocity, stress while raising natural frequency. hicker attenuate displacement velocity responses increase stress frequency. optimize performance, applied multi-objective genetic algorithm adjustable mid-span support under stochastic excitation.

Compared baseline design, Pareto-optimal solutions yield maximum upshift 28.98% fundamental frequency, while quare (RMS) displacement, velocity stress curtailed 24.83%, 24.54% 20.55%, respectively.

These findings provide quantitative benchmark sizing vibration-mitigation strategies future nuclear primary piping systems.

Keywords

tochasti vibration; uclear power plant; traight Fluid-Conveying esign

method

ology

1. Introduction

Nuclear power plant frequently exposed strong vibration influences. primary sources vibration primary piping systems originate mechanical excitation generated during nuclear reactions self-excited vibration fluid within pipes.

Under operating conditions, vibrations display tochastic characteristics.

Piping systems critical channels transporting energy media nuclear power plant performance directly impacts plant's operational efficiency.

Straight pipes, common component nuclear power plant piping systems, operate continuously under tochasti vibration conditions.

Severe vibrations generate large cyclic stresses, displacements, velocity responses. intensify bidirectional fluid-structure interaction, severely affecting pipe's vibration characteristics.

These effects cause fatigue damage, loosening wear, leaks failures serious threat reliability piping system overall safety plant. reason studying vibration response straight nuclear power transmission pipelines under stochasti excitation essential. provides basis developing effective design

method

improve reliability safety Regarding vibration response flow-conveying pipelines, focused vibration control technology pipelines.

Their research revealed pipelines susceptible vibration induced multiple factors, including fluid self-excitation, support structures, operating environments.

Severe vibrations generate variable dynamic stresses significant deformations pipelines, ultimately leading fatigue failure.

Effectively mitigating impact pipeline vibrations therefore crucial ensuring safety reliability pipelines. instance,Fan established vibration equation fluid-structure coupled pipelines based Hamilton's principle. conducted numerical

analysis

investigate influence patterns support stiffness position frequency, modal functions, critical velocity

pipeline Bozyigit conducted vibration

analysis

pipelines under various boundary conditions using differential transformation

method

Adomian decomposition method. examined natural frequencies, vibration modes, transient instability velocities pipelines under different support configurations. addition, Huang employed Galiot-Kim

method

analyze variation patterns natural frequencies simply supported straight pipelines under different boundary conditions (mass, stiffness, length, velocity).

conclusion

applies pipelines nuclear power plants other industrial fields.

Added that,Tan [5-6] employed Timoshenko model establish fluid-structure interaction model.

Using Finite Element

Method

(FEM), analyzed influence fluid structural parameters pipeline stability vibration response.

Moreover, employed finite element

method

establish Timoshenko dynamic equations fluid-conveying pipelines. investigated effects fluid velocity pipeline's natural frequency standard deviation dynamic response, calculated displacement velocity responses pipeline. addition, Sazesh examined cantilevered pipelines under distributed stochastic excitation, focusing response spectral density variance change velocity, while discussing pipeline flutter speed frequency.

Furthermore, Yamashita [9-10] investigated self-excited vibrations cantilevered pipes elastic supports masses. discovered hybrid modal self-excited vibrations caused Hopf-Hopf interactions within specific parameter ranges.

Moreover, Abdelbaki established mechanical-theory-based model straight pipes solved numerically. analyzed influence patterns different support configurations, support stiffness, structural parameters vibration response.

Furthermore, investigated elbows pipes without harmonic excitation, showing excitation structure parameters affect vibration properties. terms vibration mitigation, focuses latest research advances pipeline vibration control. examines application active passive control technologies various pipeline types engineering practice, addressing factors different geometric structures, self-excited vibrations caused fluid-structure interaction effects, complex support restraint configurations. paper discusses emerging technologies machine learning pipeline control designed tuned dampers nuclear power plant pipelines under stochastic seismic loads. reduced maximum seismic acceleration, Mean-Square (RMS) acceleration, spectral acceleration responses respectively.

Added that, Pourmohammadi constructed nonlinear energy absorber attaching sliding oscillator pipeline. nonlinear absorber effectively prevents pipeline resonance under range velocities external excitation vibration conditions Farid employed Hamilton's principle pipelines under moving loads harmonic search algorithm optimize displacement responses under varying lengths, thicknesses, velocities. addition, proposed design

method

prevent pipeline resonance adjusting fixed clamps.

Their research findings revealed modifying stiffness position clamps effectively prevent pipeline's natural frequency falling within frequency range generated mechanical operation.

Moreover, employed transfer matrix

method

establish mathematical model multi-pump pipeline system. applying chaotic swarm optimization algorithm, obtained optimal solution positioning pipeline fixtures, reducing pipeline vibration response 36.4% compared original position.

Furthermore, Colombo simulated composite pipes under actual operational conditions designed parameters thickness winding angle reduce fatigue damage. addition, Dominik addressed strong lateral vibration straight pipes critical velocities designing novel instability controller, validated through numerical simulations.

Finally, Zadkarami proposed pipeline diagnosis

method

using data-driven neural networks. collecting training real-time pressure signals using neural network models,

method

achieved accuracy. Regarding nuclear power plant piping systems, investigated effectiveness tuned dampers under various seismic

conditions constructing full-scale numerical model nuclear piping system, evaluated their impact seismic vulnerability curve piping system.

Moreover, addressed issue performance degradation flow-carrying piping systems under seismic loads employing piping system numerical indicator reflect deterioration level nuclear piping addition, developed fluid-structure interaction mathematical model water pipes nuclear reactor insulation layers. analyzed influence patterns temperature stress, experimentally validated model's accuracy.

Added that, proposed novel reliability

method

nuclear pipelines, coupling failure physics models maintenance performance

analysis

models monitor real-time pipeline condition pipeline defect proposed real-time unmeasured dynamic response prediction

method

nuclear power plant pressure pipes directly measuring limited physical responses predict unmeasurable inaccessible dynamic responses acceleration, velocity, proposed novel vibration reduction design

method

integrating BP-ANN NSGA-II algorithm optimizing dynamic response flow-carrying pipes nuclear power plants. approach derived optimal design parameters pipes, effectively reducing vibration-induced failures, experimentally validated correctness method.

Furthermore, Tomarov investigated characteristics erosion-corrosion interaction between single-phase two-phase fluids metals nuclear power plant working circuits. proposed

method

based kinetic migration theory identify regions localized maximum erosion-corrosion thinning piping components equipment within nuclear power plant addition,Added that, employed virtual crack distribution model simulating characteristics crack distribution actual operation assess probability failure nuclear power plant piping. determined likelihood piping failure analyzing crack depth distribution, crack density, level detection Dacosta employed large simulation analyze fluid characteristics mixing points. developed detection

method

fatigue assessment mixing points nuclear transport pipelines, targeting fatigue-prone locations welded joints stress concentration addition, Cancemi established finite element simulation model nuclear pipelines. analyzed simulation

results

using neural networks ARIMA models predict remaining service nuclear established experimental platform detecting leaks nuclear pipelines, identifying leaks locating faults analyzing effects aperture internal pressure time-frequency characteristics induced acoustic proposed novel online structural integrity assessment system nuclear power plant pipelines. system utilizes thermoelectric potential failure assessment mapping technology enable real-time monitoring actual pipeline aging conditions transient operational developed online detection system nuclear pipelines based machine learning-driven fracture angle diagnosis, establishing comprehensive early warning-localization-angle monitoring diagnostic framework.

Finally, Allah application machine learning learning technologies corrosion crack detection within nuclear power plant transport pipelines. approach emerged promising solution enhancing accuracy efficiency corrosion crack detection methods. domestic international research pipelines primarily focused self-excited harmonic-excited fluid within pipes.

Meanwhile, studies nuclear pipelines concentrated materials, corrosion, leakage, fatigue, performance monitoring.

However, under actual operating conditions, nuclear power pipelines typically function stochastic vibration environments.

Little research conducted impact support position changes pipeline vibration characteristics optimized design nuclear pipeline structural parameters.

Therefore, paper addresses analyz natural frequency, displacement, velocity, stress straight nuclear pipeline fixed movable fixed support under different structural parameters. objective reduce pipeline displacement, velocity,

stress responses while raising natural frequency Based these findings, propose design

methodology

nuclear pipelines reflect actual operating conditions helps prevent damage failure caused design flaws.

2. Establishment

Solution Straight Mathematical Models Establishment mathematical model stochastic vibration Stochastic vibrations generated nuclear reaction transmitted pipeline through fixed supports.

Based position movable fixed support pipeline divided equal sections where denotes position movable fixed support, where illustrated Figure tochastic vibration is transmitted to the pipeline through rigid supports.

Physical flow-conducting straight under stochastic vibration mathematical model vibration straight flow-carrying under stochastic vibration established using tochastic vibration theory, bidirectional fluid-structure interaction theory, finite element method. vibration equation follows:

} { } ] { [ } ] { [ } ] { [ X b a K a C a M = + + ( 1 )

where represents overall matrix pipeline, denotes overall damping matrix, indicates overall stiffness matrix, refers displacement response vector, tochastic vibration intensity vector indicating location intensity stochastic vibration, represents stochastic vibration acceleration.

Pipelines typically analyzed using Timoshenko model, which pipeline treated thin-walled slender while accounting shear strain effects moment inertia. illustrated Figure represents lateral displacement element, denotes cross-sectional rotation angle, indicates longitudinal displacement element.

Timoshenko lement According finite element assumptions, displacement field straight expressed follows Moreover, strain energy straight under stochastic vibration denoted

¶ ¶ + ¶ ¶ =

kinetic energy straight flow-conveying under stochastic vibration modelled follows

• ¶ ¶ + + ¶ ¶ + =

According Hamilton's principle, obtain:

1 = - ò d t W T t

denotes shape function matrix longitudinal displacement pipeline, represents shape function matrix lateral displacement pipeline, indicates shape function matrix rotational displacement pipeline cross-section, represents vector degrees freedom nodes straight element, denotes length element, elastic modulus, indicates moment inertia pipeline cross-section, represents cross-sectional pipeline element, longitudinal position coordinate element, shear modulus, denotes shear coefficient, represents shear strain, length fluid element, highlights fluid velocity, fluid pressure pipe, length element, denotes volumetric element, volumetric fluid element, represents sectional moment inertia element, sectional moment inertia fluid element. study assumes steady, incompressible fluid accounts bidirectional fluid-structure interaction between pipeline fluid. ubstituting damping matrix, stiffness matrix, matrix straight element obtained respectively:

[ ] [ ] [ ] [ ] [ ] v d x m B B B B C f L T x x T e i ò - =

• • =
• • • =

where denotes damping matrix straight element, represents stiffness matrix straight element, indicates matrix straight element, subscript represents partial differentiation respect Using entire pipeline assembled applying geometric constraints element nodes enforcing boundary conditions process yields overall matrix damping matrix stiffness matrix Solving stochastic vibration mathematical models modal

method

employed determine pipeline natural frequencies. Introducing state vector:

Substitute state vector equation transformed follows

M    = ú û

where

é = K

î í ì = 0

• = 0 K

rewritten follows natural frequency pipeline depends solely inherent structural properties independent external excitation. considering pipeline's self-excited vibration absence external excitation, simplified

= + + a K a C a M ( 12 )

Similarly, simplifie

= + B w w A ( 13 )

solution following where denotes eigenvector represents eigenvalue. eigenvalues known

é - - =

According complex modal theory, equation solutions, where number degrees freedom system These solutions complex eigenvalues =1,2, positive imaginary parts imaginary corresponds natural frequency conduit. irtual excitation

method

applied calculate pipeline stress, displacement, velocity responses. practice flow-carrying pipelines nuclear power plant operations experience intensive vibrations difficult characterize specific functions These vibrations exhibi distinct randomness.

However, during collection vibration signals actual sites, found their probability distribution follow recognizable statistical patterns, commonly normal distributions.

Based observation stochastic vibration considered paper modeled stationary Gaussian white noise acceleration conform normal distribution.

Based (10), (11), vibration equation straight flow-carrying under stochastic vibration expressed follows right characteristic matrices satisfy following equations:

( ) 0 a = + V B A T l ( 18 )

Based modal

analysis

method, following decoupling

results

obtained:

Substituting multiplying sides apply orthogonality right characteristic matrices transform following equations where self-power spectral density stochastic vibration whose variance given

• ¥ =

vibration response -conveying pipelines under stochastic vibration solved using virtual excitation method, where stochastic vibration defined follows solution expressed follows Substituting yields:

a s - = = å å

(25), displacement velocity responses pipeline under stochastic vibration obtained. spectral matrix response derived follows

( ) x y y y y , × = * t S w (26)

expression calculating velocity response

• ¥ = (27)

expression calculating displacement response

• ¥ = (28)

Finally, expression calculating stress response

• ¥ = (29)

3. Analysis

Natural Frequency Straight Pipes Fluid Based actual operating parameters primary pipeline nuclear power plant, shown Parameter Operating Value

length/m Elastic Modulus/Pa Pipeline Density/(kg Poisson ratio( Liquid density/(kg Average fluid pressure/Pa Average fluid velocity/(m other parameters constant, single-factor

method

conducted examine influence movable support position, length, inner diameter, thickness natural frequency, illustrated Movable support position length inner diameter

thickness Effect ipeline tructural arameters atural requency shown Figure natural frequency movable support first increases decreases shifts along pipeline, reaching maximum Increasing inner diameter thickness raises natural frequency.

Among positions, movable support exhibits largest increase, while shows smallest increase gradual contrast, increasing length reduces natural frequency. again, natural frequency highest movable support, exhibiting sharpest decrease, while shows slowest decline Vibration Response

Analysis

Flow-Carrying Straight Pipes other parameters constant, single-factor

method

employed. random vibration modeled Gaussian white noise variance mean.

Pipeline isplacement esponse nalysis effects movable support position, length, inner diameter, thickness displacement response analyzed shown Figure length inner diameter

thickness Effect ipeline tructural arameters isplacement esponse shown Figure displacement outlet increases length. movable support displacement reaches maximum exhibits steepest increase.

Conversely, positioning support

results

minimum displacement slowest growth Increasing inner diameter thickness reduces displacement outlet. support square displacement largest decreases sharply, while smallest, slowest decrease.

Pipeline esponse nalysis effects movable support position, length, inner diameter, thickness velocity response analyzed shown Figure length inner diameter

thickness Effect pipeline structural parameters velocity response shown Figure velocity outlet increases length. movable support velocity reaches maximum exhibits steepest increase.

Conversely, positioning movable support

results

minimum velocity slowest increase. Increasing inner diameter thickness reduces velocity outlet. velocity outlet highest support exhibiting pronounced decrease, while lowest being showing slowest decrease.

Pipeline tress esponse nalysis influence movable support positions, length, inner diameter, thickness stress response analyzed illustrated Figure length inner diameter

thickness Influence ipeline tructural arameters tress esponse shown Figure stress outlet decreases increasing length inner diameter. movable support stress highest decreases sharply contrast, support stress lowest decrease gradually Increasing thickness leads higher stress outlet. support stress highest increase steeply, while lowest rises slowly

Method

Straight Piping

Analysis

Section shows pipeline structural parameters support locations significantly affect vibration response nuclear power flow-carrying straight pipes under stochastic vibration.

However, single-factor

analysis

cannot capture interactions among these parameters, whereas actual pipeline vibration responses

result

combined effects multiple factors Therefore, necessary design primary structural parameters straight flow-carrying pipes while simultaneously considering movable support locations minimize displacement, velocity, stress responses, providing practical design

methodology

pipelines under stochastic vibration conditions. Design mathematical models Designing structural parameters nuclear power flow-carrying pipelines under stochastic vibration multi-objective, multi-constraint, multi-extremum problem. address multi-objective genetic algorithm selected. multi-objective genetic algorithm design problem employs following mathematical model: where V-min denotes scenario where subfunctions multi-objective function reach their optimal solutions.

Based engineering practice

analysis

Section three primary structural parameters straight selected design variables: -pipe length, -pipe inner diameter, -pipe thickness i.e., objective minimize displacement, velocity, stress responses pipeline under stochastic vibration determined Section calculation formulas these responses given (27), (28), (29), respectively:

Moreover, constraint design variables defined follows

Design

analysis

Based mathematical model, objective function, constraint commonly parallel selection

method

selected solve optimization problem. straight flow-supporting fixed movable fixed support representative example.

Optimal solution displacement response Optimal solution stress response Variation ptimal olution bjective unction shown Figures displacement velocity responses which exhibit largely similar trends, converge their values around generation, yielding optimal solution. stress response converges earlier, around generation, providing optimal solution. omparison pipeline structural parameters before after optimization presented Table Pipeline Structural Parameters length Inner diameter thickness

Initial value 1.0000 0.067 0.0045

values displacement, velocity, stress response outlet fixed-supported, movable-supported flow-through straight under stochastic vibration conditions before after design modification illustrated Figures

displacement outlet before after optimization velocity outlet before after optimization stress outlet before after optimization Natural frequency piping before after optimization Figures after optimization, maximum displacement, velocity, stress outlet decreased 24.83%, 24.54%, 20.55%, respectively. ecalculating pipe's natural frequency optimized parameters revealed maximum increase 28.98%, illustrated Figure These

results

indicate optimizing pipeline structural parameters effectively reduces vibration response under stochastic vibration while

simultaneously increasing pipeline's natural frequency. significantly enhances pipeline safety reliability.

Pipeline design process

summary

vibration response pipelines significantly affected structural parameters support locations.

Traditional design

methods

insufficient nuclear power pipelines under actual operating conditions.

Therefore, design process straight pipelines under stochastic vibration proposed shown Figure Design process nuclear power flow-through straight pipes under stochastic vibration

6. Conclusion

mathematical model design

methodology

established vibration response nuclear power plant flow-carrying pipelines under stochastic vibration.

Using complex modal

analysis

virtual excitation method, pipeline's natural frequencies, displacements, velocities, stress responses determined. (2)Research findings support position length, inner diameter, thickness significantly affect pipe's natural frequency, displacement, velocity, stress response.

Specifically: movable support positioned natural frequency stress response highest, while displacement velocity responses lowest movable support positioned natural frequency stress response lowest, while displacement velocity responses highest length increases, natural frequency stress response decrease, while displacement velocity responses increase inner diameter increases, natural frequency increases however, displacement, velocity, stress responses decrease thickness increases, natural frequency stress response increase, displacement velocity responses decrease.

Using multi-objective genetic algorithm, structural parameters straight flow-conveying pipeline fixed-end support movable support optimized under stochastic vibration conditions.

After optimization, maximum displacement, velocity, stress pipeline outlet decreased 24.83%, 24.54%, 20.55%, respectively, while maximum natural frequency increased 28.98%.

Building these results, design

methodology

straight flow-conveying pipes under stochastic vibration proposed enhancing performance reliability nuclear power flow-conveying pipelines high-vibration environments. (4)Building research presented paper, subsequent studies delve nonlinear dynamics, fatigue prediction, multimodal vibration reduction design nuclear power plant flow-carrying pipelines.

Acknowledgments

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