IPS NASU DYNAMICAL ANALYSIS AND ALLOWABLE VIBRATION DETERMINATION FOR THE PIPING SYSTEMS. G.S. Pisarenko Institute for Problems of Strength of National

IPS NASUIPS NASU

DYNAMICAL ANALYSIS AND ALLOWABLE VIBRATION DETERMINATION FOR THE PIPING

SYSTEMS.

DYNAMICAL ANALYSIS AND ALLOWABLE VIBRATION DETERMINATION FOR THE PIPING

SYSTEMS.

G.S. Pisarenko Institute for Problems of Strengthof National Academy of Science of Ukraine

Kiev, Ukraine

G.S. Pisarenko Institute for Problems of Strengthof National Academy of Science of Ukraine

Kiev, Ukraine

IPS NASUIPS NASUSoftware complex

«3D PipeMaster»Software complex

«3D PipeMaster» Method of calculation of piping at harmonical vibrationsMethod of calculation of piping at harmonical vibrations

Modeling of dynamical behavior of pipe bend as the Modeling of dynamical behavior of pipe bend as the beam as well as the shell beam as well as the shell

The abilities of the complex for vibrodiagnostics The abilities of the complex for vibrodiagnostics

Accident of the oil Accident of the oil pipeline pipeline

IPS NASUIPS NASU«3D PipeMaster»

Harmonical analysis«3D PipeMaster»

Harmonical analysisDynamic stiffness methodDynamic stiffness method

x

y

dx

X0X1

01 ),( XdxAX

stiffness matrixy

with method of initial parameterswith method of initial parameters

x

X10

2 … n-1 nX1

1 X20 X2

1 Xn-10 Xn

0Xn-11 Xn

1

1

;11

0

ii XX ;)( 0

01

XAX n

n

iini dxAA

11,)(

The sweeping procedure

IPS NASUIPS NASU

The inertial term

«3D PipeMaster» Harmonical analysis

«3D PipeMaster» Harmonical analysis

Dynamic stiffness methodDynamic stiffness method

02

4

4

yz

yW

EI

F

dx

Wd z

y

dx

dW

z

zz

EI

K

dx

d

y

z Qdx

dK

the equations of motion at transversal vibrations the equations of motion at transversal vibrations

- frequency of vibration- frequency of vibration

the equations of the method of initial parameters: the equations of the method of initial parameters:

xkYkEI

QxkY

kEI

KxkY

kxkYWW y

yz

y

yyz

z

yy

z

yyy 433221000

0

xkYWkxkYkEI

QxkY

kEI

KxkY yyyy

yz

yy

yz

zyzz 43221 0

00

0

xkYEIkxkYEIkWxkYk

QxkYKK yzyzyzyyy

y

yyzz 43

221 00

0

0

xkYkKxkYEIkxkYEIkWxkYQQ yyzyzyzyzyyyyy 432

23

1 0000

zy EI

Fk

24



Harmonical analysisThe algorithms for branched and curvelinear elementsThe algorithms for branched and curvelinear elements

1

1

2

2

3

3

4

4 5

1

;,1,e

izb

iz

;sincos,1, ieii

eiy

biy

;sincos ,1 ie

iyiei

bi

;sincos,1 ieii

eiy

biy, UWW

;1eiz,

biz, WW

.sincos1 ieiy,i

ei

bi WUU

the conditions in the junctionsthe conditions in the junctions

equations for pipe bend equations for pipe bend

The matrix of the turning element

;)( 11

0ii XBX

i

;)( 001 XCX n

;)(,)( 11

1

in

n

iini BdxAC

...321 WWW

...321 0M

0Q

1

2

3

m



Harmonical analysisMethod of the breaking of displacements for the Method of the breaking of displacements for the

determination of the natural frequencies and forms determination of the natural frequencies and forms

0

0

0,

11,

QQ

orQQiy

iyx

i-1 iXi-1

0 Xi0Xi-1

1 Xn1

y 11,0 i

yiy, WW 1

1,0 i

yiy,y WWW

the criteria of the determination of the natural frequencythe criteria of the determination of the natural frequency

- natural frequency - natural frequency

0)(yW

The example of the graph for T –like frame The example of the graph for T –like frame

)(yW



Harmonical analysisMethod of the breaking of displacements continuityMethod of the breaking of displacements continuity

The role of the estimator is essential The role of the estimator is essential !!!!!!

The additional frequency can be noticed only at very small step of frequency. The additional frequency can be noticed only at very small step of frequency.



Harmonical analysisMethod of the breaking of displacements continuityMethod of the breaking of displacements continuity

The examples of finding the natural frequencies and forms for T-like frameThe examples of finding the natural frequencies and forms for T-like frame

=148 с-1=148 с-1 =212.4 с-1=212.4 с-1 =214.4 с-1=214.4 с-1

The additional form The additional form of vibrationof vibration !!! !!!

-1

-1

1

0.03

-1 -1

-1

1

1

The forms given in the handbooksThe forms given in the handbooks



Harmonical analysisMethod of the breaking of displacementsMethod of the breaking of displacements

modeling of curvilinear elementmodeling of curvilinear elementExample: frequencies of the circular ringExample: frequencies of the circular ring

Е = 2∙106 МПа; G = 8∙105 МПа; = 0.3; = 8000 кг/м3;В0 = 2 м; R = 0.1 м

Е = 2∙106 МПа; G = 8∙105 МПа; = 0.3; = 8000 кг/м3;В0 = 2 м; R = 0.1 м

n = 2 n = 3 n = 4 n = 5

Vibration in the plane of circular ring

theoretical 167.7051 474.3416 909.5086 1470.8710

Our results 167.569 473.857 908.4868 1469.146

Out-of-plane vibration of circular ring

теоретическое 163.6634 468.5213 902.8939 1463.8510

наши результаты 163.36 467.371 900.391 1459.662

Kang K.J., Bert C.W. and Striz A.G.Kang K.J., Bert C.W. and Striz A.G.Vibration and buckling analysis of circular Vibration and buckling analysis of circular arches using DQMarches using DQM// Computers and Structures. – 1996. –V.60, // Computers and Structures. – 1996. –V.60, №№1. 1. – pp. 49-57.– pp. 49-57.

,

11

2

222

40

nnn

FBEI z

2n

vibrations in planevibrations in plane

Out-of-planeOut-of-plane

,

1

1402

22

FB

GI

EI

GIn

nn y

y

кр

2n



Harmonical analysisMethod of the breaking of displacementsMethod of the breaking of displacementsmodeling of curvilinear elementmodeling of curvilinear elementExample: frequencies of the circular arcExample: frequencies of the circular arc

1. In-plane vibrations Austin W.J. and Veletsos A.S. Free vibration of arches flexible in shear // J. Engng Mech. ASCE. – 1973. – V.99. – pp. 735-753.2. Out-of-plane Ojalvo U. Coupled twisting-bending vibrations of incomplete elastic rings // Int. J. mech. Sci. – 1962. – V.4. – pp. 53-72.

1. In-plane vibrations Austin W.J. and Veletsos A.S. Free vibration of arches flexible in shear // J. Engng Mech. ASCE. – 1973. – V.99. – pp. 735-753.2. Out-of-plane Ojalvo U. Coupled twisting-bending vibrations of incomplete elastic rings // Int. J. mech. Sci. – 1962. – V.4. – pp. 53-72.



Harmonical analysisAdvantagesAdvantages

1. The strict analytical solutions are used. 2. The continuity is provided at transition from static to dynamic3. The infinite number of natural frequencies can be obtained for finite number of elements.4. The method of sweeping allows to speed up the calculation.5. Analytical accuracy of modeling of curved element is attained.6. Any complex spatial multibranched piping system can be treated.7. The vibration direction (modes) of interest can be separated8. The influence of the subjective factors are excluded (the breaking out on the elements)

IPS NASUIPS NASU

Dynamical model of pipe bend as the beam as well as the shell Dynamical model of pipe bend as the beam as well as the shell

,02

4

4

WEIF

Kdx

Wd d

A

C

B

D

A

C

B

D

zKzK

1d

0d

B

O

R

bendpipeforxPfK

pipestraightforK

д ,,,,

1

BR

BtR2

- flexibility parameter- flexibility parameter

- parameter of curvature - parameter of curvature

The curved beam element is strict but pipe bend have the increased flexibility!The curved beam element is strict but pipe bend have the increased flexibility!

Depends from the frequency !Depends from the frequency !

Physical equation is corrected Physical equation is corrected EIMK

dxd

Equation of the transversal vibration with accounting of increased flexibility:Equation of the transversal vibration with accounting of increased flexibility:

IPS NASUIPS NASU

Equation for bend as a shellEquation for bend as a shellr

R

O

B

O1

t

x

y

z

v u

w Equilibrium equations:Equilibrium equations:

0sin1

2

2

0

tw

hB

Nx

QQ

RR

Nxx

0cos1

2

2

0

tv

hB

N

xL

R

QN

Rx

0sin

2

2

0

t

uh

B

Q

x

NL xx

01

x

MM

RQ x

01

x

MM

RQ xx

x

HN

HNx

12H

L

HM

HM x

21 H

M x

Physical equationsPhysical equations

Determination of the flexibility of the pipe bend


IPS NASUIPS NASU

0

sincos

B

wv

x

u

Rwv

R

1

2

2

xw

22

2

2

1Rww

R

2

2

xw

xw

Rxv

R

222

deformations deformations

curvatures curvatures

Geometrical equations:Geometrical equations:



The simplifications:The simplifications: semimomentless Vlasov’s theory: semimomentless Vlasov’s theory: 0,...,0

v

wxv

Ru

,...,

geomtrical characteristics: geomtrical characteristics: ,62

BtR 0

BR

restrictions on the wave length in the axial direction

restrictions on the wave length in the axial direction

2

2

2

2

v

x

v

0sincos21

4

4

2

2

2

2

002

2

2

3

2

2

4

4

vv

th

N

BB

N

x

NR

QQ

Rxxx

IPS NASUIPS NASU



Solution for the cylindrical shellSolution for the cylindrical shell

0112

11

,sinsin,,,

6

2

2

22422

2

2

0

nIV

n

n

VRhnn

nnER

V

tnxVtxvB

ASR

Rhnn

nnRE

m

m

,

1121

1 2

2

2

222

2

4

222

600

800

1000

1200

1400

1600

0,2 1,1 1,2 2,2 3,2

мода колебаний (m, n)

част

ота,

Гц

experiment

FEA [Salley and Pan]

our results

Salley L. and Pan J.A study of the modal characteristics of curved pipes // Applied Acoustics. – 2002. – V.63. – pp. 189-202.

IPS NASUIPS NASU



Решение для гибаРешение для гиба

ERB

txVxVRtxv 032 ,sin...3cos,2sin,,,

,2

31, 2 xV

xkxK д

The sought for solutionThe sought for solution::

The resulting equations:The resulting equations:

nIV

nnnnnnnnnnn faVVaVaVaVaVa 1,51,42,32,2,1

3,161,1

2,12144,1

22224,1

22,1,1

,122

,1,1

nnnAnnaBnnaa

nAaBnnaa

nnn

nnn

;133 2,2 nnnAa n

;1 22,4,4 h

RRaa nn

;1

11223

1,4

nnnn

a nn

;)1()1( 22

220

24

hB

RA

;112 42

22 R

hR

a

;133 2,3 nnnAa n

;1 22,5,5 h

RRaa nn

;1

11223

,5

nnnn

a nn

;3,0

;2,722

nf

nxAkf

n

Eh

RB

2

242 )1(12

IPS NASUIPS NASU



- The coefficient of flexibility at harmonical vibrations

B

BABBABAK

2

10607241167211672 2

22 )1( A

0

5

10

15

20

25

30

0 10 20 30 40 50 60

A=1A=3A=6A=10A=15A=30

K д

B

EhR

B 2

242 )1(12 BhR2

nz

дIV

n VEI

FKV

22 ),( AssumeAssume::

45138.28

дKA

BB

ifif

then we obtain then we obtain ::

3,161

,

2,12144

,1

22224,1

,1,1

,1

22,1,1

nnnAnna

aa

nAa

Bnnaa

n

nn

n

nn

Results:Results:

IPS NASUIPS NASU

200

600

1000

1400

1800

2200

2600

3000

R,2s R,2a 1,2s 1,2a 3,2s 3,2a 1,1a

мода колебаний (m, n)

част

ота

, Гц

experiment

FEA [Salley and Pan]

our results withdynamical КK=1

the Saint-Venant(static) solution

L. Salley and J. Pan. A study of the modal characteristics of curved pipes // Applied Acoustics. – 2002. – V.63. – pp. 189-202.

Е = 2.07∙106 МПа; = 0.3; = 8000 кг/м3; R = 0.0806 м; h = 0.00711 м;В = 0.457 м l=0.2 м

l

l

Rh

B



IPS NASUIPS NASU

P

A B

l

W

х

thtglPl

M

820 4

42

2

1

EI

lF

tPtP cos0

-4

-3

-2

-1

0

1

2

3

4

0 50 100 150 200 250 300

M(l /2)

частота, рад/c

Е = 2∙106 МПа; G = 8∙105 МПа; = 0.3; = 8000 кг/м3; l = 5 м; R = 0.1 м; h = 0.005 м.

1. The graph of bending moment in the central point of supported-supported beam

srad136

1.25

2. Restoration of the outer force from the known displacements in arbitrary point

P0, H

-4.E+07

-3.E+07

-2.E+07

-1.E+07

0.E+00

1.E+07

2.E+07

3.E+07

4.E+07

5.E+07

6.E+07

0 500 1000 1500

частота, рад/c

1

10 P10 W

Abilities of «3D PipeMaster» for vibrodiagnostics

Abilities of «3D PipeMaster» for vibrodiagnostics

IPS NASUIPS NASUAbilities of «3D PipeMaster» for

vibrodiagnosticsAbilities of «3D PipeMaster» for

vibrodiagnosticsThe problems of vibrodiagnosticsThe problems of vibrodiagnostics1. The points of application of the outer forces, their directions and frequencies are unknown. 2. The gauges can measure the displacements of pipe points, their velocities and accelerations3. The number of gauges is finite.

The functions of the calculation softwareThe functions of the calculation software1. The correct determination of the dynamical characteristics. 2. Correct modeling of the piping behavior when the correct measurement data are provided. 3. The best possible assessment of the behavior with restricted input data.4. The best possible assessment of the dynamical stresses based on the incomplete measurements



vibrodiagnostics

yy WF ,

A B

ll 20

х

yQWM ,, yQWM ,,

tFtFy sin0Е = 2.0689∙106 МПа; μ= 0.3;= 7836.6 кг/м3; l = 6.096 м; Δl=0.3048м; R = 0.05715 м;t = 0.0188 м. 11.66 Гц, 37.65 Гц, 78.18 Гц.

1 23

1.1. Input data are the results of excitation of beam by harmonical force applied at its Input data are the results of excitation of beam by harmonical force applied at its center. The calculated values of transverse forces, bending moment, displacements center. The calculated values of transverse forces, bending moment, displacements in 21 points are recordedin 21 points are recorded. . This is so called This is so called ««real casereal case».».

2.2. The system (beam) is loaded by The system (beam) is loaded by ««the realthe real» » displacements in a few (or one) pointsdisplacements in a few (or one) points, , the moments and displacements are calculatedthe moments and displacements are calculated..

3.3. The calculated in 2 results are compared with The calculated in 2 results are compared with ««real datareal data».».

The frequency of outer force is given but the point of its application is unknown. The gauges measure the displacements

The frequency of outer force is given but the point of its application is unknown. The gauges measure the displacements



vibrodiagnostics

A B

х

l3 l3

-0.0008

-0.0007

-0.0006

-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

0

0.0001

0 2 4 6 8 10 12 14 16 18 20

Длина, deltaL

W

-600

-400

-200

0

200

400

600

M

W восст. W "реал." M восст. M "реал."

=21 Гц=21 Гц

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 5 10 15 20 25

Длина, deltaL

W

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

M


=8 Гц=8 Гц

2 points of measurements2 points of measurements



vibrodiagnostics

=100 Гц=100 Гц=80 Гц=80 Гц

=60 Гц=60 Гц-0.00015

-0.0001

-0.00005

0

0.00005

0.0001

0 2 4 6 8 10 12 14 16 18 20

Длина, deltaL

W

-600

-400

-200

0

200

400

600

M


-0.0008

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

0.0008

0 2 4 6 8 10 12 14 16 18 20

Длина,deltaL

W

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

M


-0.0004

-0.0003

-0.0002

-0.0001

0

0.0001

0.0002

0.0003

0.0004

0.0005

0 2 4 6 8 10 12 14 16 18 20

Длина, deltaL

W

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

M




vibrodiagnostics

A B

х

l l

=140 Гц=140 Гц=60 Гц=60 Гц


-0.00014

-0.00012

-0.0001

-0.00008

-0.00006

-0.00004

-0.00002

0

0.00002

0 2 4 6 8 10 12 14 16 18 20

Длина, deltaL

W

-600

-400

-200

0

200

400

600

M


-0.00006

-0.00004

-0.00002

0

0.00002

0.00004

0.00006

0 2 4 6 8 10 12 14 16 18 20

Длина, deltaL

W

-600

-400

-200

0

200

400

600

M




vibrodiagnostics

=100 Гц=100 Гц=60 Гц=60 Гц

A B

х

l3 l3 l2 l2


-0.00014

-0.00012

-0.0001

-0.00008

-0.00006

-0.00004

-0.00002

0

0.00002

0 5 10 15 20

Длина, deltaL

W

-600

-400

-200

0

200

400

600

M


-0.00014

-0.00012

-0.0001

-0.00008

-0.00006

-0.00004

-0.00002

0

0.00002

0.00004

0.00006

0.00008

0 2 4 6 8 10 12 14 16 18 20

Длина, deltaL

W

-600

-500

-400

-300

-200

-100

0

100

200

300

400

500

M




vibrodiagnosticsAll measurements in all points are used All measurements in all points are used

Complete coincidenceComplete coincidence

Conclusions from modeling:Conclusions from modeling:

1. 1. To evaluate stresses the most importance have the To evaluate stresses the most importance have the proximity of the points of measurements to the point of proximity of the points of measurements to the point of the force applicationthe force application..2. 2. The accuracy grows with the number of the points of The accuracy grows with the number of the points of measurement measurement 3. 3. The accuracy nonmonotically decrease with the The accuracy nonmonotically decrease with the frequency of the excitationfrequency of the excitation

1

,2

form

formMAXt

MAX

C

ECW



vibrodiagnosticsDetermination of the maximal stresses based on the measurements of velocities Determination of the maximal stresses based on the measurements of velocities

kt

k

MAX

Lx

kAW

EI

mk

ERkL

AEIdx

WdMR

IM

*sin

)(

,

24

22

2

2

2

For simply supported beam:For simply supported beam:

IFR

E

IE

ERm

W MAXt

MAX

2

For a thin walled pipe:

for a solid circular beam:

For the real complex piping systems:

For a thin walled pipe:

for a solid circular beam:

For the real complex piping systems:

EW MAXt

MAX 2

EW MAXt

MAX 4

dynamic susceptibility coefficient

dynamic susceptibility coefficient



vibrodiagnosticsExamples of the piping configurationExamples of the piping configuration



vibrodiagnosticsDetermination of the maximal stresses based on the measurements of velocitiesDetermination of the maximal stresses based on the measurements of velocities

Е = 2.06843∙106 МПа;= 7834 кг/м3; l = 18 м;R = 0.1 м; t = 0.01 м.

J. C. Wachel, Scott J. Morton, Kenneth E. Atkins. Piping vibration analysis

msPа

10*98.5 7 MAXt

MAX

W

Theoretical value:Theoretical value:



vibrodiagnosticsDetermination of the maximal stresses based on the measurements of velocitiesDetermination of the maximal stresses based on the measurements of velocities

When the exciting frequency exceeds the first natural frequency the correlation between the vibrovelocity and maximal stresses is good

Е = 2.0689∙106 МПа; ρ=7836.6 кг/м3; R = 0.05715 м; t = 0.0188 mFor parametersFor parameters

Theoretical value 11.66 hertzTheoretical value 11.66 hertz m

Pа10*5.69 7 s

W MAXt

MAX

ω, Гц 2 8 21 40 60 80

3.08E+08 7.78E+07 2.87E+07 5.71E+07 5.12E+07 5.55E+07

obtained value 5.4 1.4 0.51 1 0.9 0.98

MAXt

MAX

W

The results of calculation:The results of calculation:

1

IPS NASUIPS NASU Conclusion Conclusion

1. Due to application of dynamical stiffness method the continuity between the static and dynamic solution is provided.2. The procedure of the breaking of the displacements in any point and in any direction allow to find all natural frequencies and forms3. In a first time in a literature the notion of dynamic coefficient of pipe bend flexibility is introduced and analytical expression for it is obtained. This allowed to perform calculation for the piping systems with a higher accuracy4. The option of determination of exciting force in some point based on given displacement or velocity in any other point of the piping allows to efficiently perform the vibrodiagnostic analysis

Documents

IPS NASU DYNAMICAL ANALYSIS AND ALLOWABLE VIBRATION DETERMINATION FOR THE PIPING SYSTEMS. G.S. Pisarenko Institute for Problems of Strength of National