10 Ukr Hungary Dynam

8/11/2019 10 Ukr Hungary Dynam

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IPS NASU

DYNAMICAL ANALYSIS AND ALLOWABLE

VIBRATION DETERMINATION FOR THE PIPING

SYSTEMS.

G.S. Pisarenko Institute for Problems of Strength

of National Academy of Science of Ukraine

Kiev, Ukraine


2/31

IPS NASUSoftware complex

3D PipeMaster

Method of calculation of piping at harmonical vibrations

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

The abilities of the complex for vibrodiagnostics

Accident of the oil

pipeline


3/31

IPS NASU3D PipeMaster

Harmonical analysisDynamic stiffness method

x

y

dx

X0X1

01 ),( XdxAX

stiffness matrixy

with method of initial parameters

x

X10

2 n-1 n

X11 X2

0 X2

1 Xn-1

0 Xn

0Xn-1

1 Xn

1

1

;11

0

ii XX ;)( 001

XAXn

n

i

ini dxAA1

1,)(

The sweeping

procedure


4/31

IPS NASU

The inertial term

3D PipeMaster

Harmonical analysisDynamic 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

- frequency of vibration

the equations of the method of initial parameters:

xkYkEI

QxkY

kEI

KxkY

kxkYWW y

yz

y

y

yz

z

y

y

z

yyy 433221000

0

xkYWkxkYkEI

QxkY

kEI

KxkY yyyy

yz

y

y

yz

z

yzz 43221 0

00

0

xkYEIkxkYEIkWxkYk

QxkYKK yzyzyzyyy

y

y

yzz 43

2

21 00

0

0

xkYkKxkYEIkxkYEIkWxkYQQ yyzyzyzyzyyyyy 432

2

3

1 0000

zy

EI

Fk

24


5/31


Harmonical analysis

The algorithms for branched and curvelinear ele

ments

1

1

2

2

3

3

4

45

1

;,1,e

iz

b

iz

;sincos,1, ie

ii

e

iy

b

iy

;sincos ,1 ie

iyi

e

i

b

i

;sincos,1 ie

ii

e

iy

b

iy, UWW

;1e

iz,

b

iz, WW

.sincos1 ie

iy,i

e

i

b

i WUU

the conditions in the junctions

equations for pipe bend

The matrix of the turning

element

;)( 11

0

ii XBXi

;)( 001 XCXn ;)(,)( 1

1

1

inn

i

ini BdxAC

...321 WWW

...321

0M

0Q

1

2

3

m


6/31


Harmonical analysis

Method of the breaking of displacements for thedetermination of the natural frequencies and forms

0

0

0,

11,

QQ

orQQ

i

y

iyxi-1 i

Xi-10 Xi0X

i-11 X

n1

y 11,0

iyi

y, WW 1

1,0

iyi

y,y WWW

the criteria of the determination of the natural frequency

- natural frequency 0)(yW

The example of the graph for T

like frame)(yW


7/31


Harmonical analysis

Method of the breaking of displacements continuity

The role of the estimator is essential !!!

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

frequency.


8/31


Harmonical analysis

Method of the breaking of displacements continuity

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

like frame

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

The additional form

of vibration !!!

-1

-1

1

0.03

-1 -1 -1

1

1

The forms given in the handbooks


9/31


Harmonical analysisMethod of the breaking of displacements

modeling of curvilinear elementExample: frequencies of the circular ring

= 2106; G = 8105;

= 0.3; = 8000 /3;

0= 2 ;R= 0.1

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

163.6634 468.5213 902.8939 1463.8510

163.36 467.371 900.391 1459.662

Kang K.J., Bert C.W. and Striz A.G.

Vibration and buckling analysis of circular

arches using DQM

// Computers and Structures.1996.V.60,1.

pp. 49-57.

,

1

12

222

4

0

n

nn

FB

EIz

2n

vibrations in plane

Out-of-plane

,

1

14

02

22

FB

GI

EI

GIn

nn y

y

2n


10/31


Harmonical analysis

Method of the breaking of displacementsmodeling of curvilinear elementExample: frequencies of the circular arc

1. In-plane vibrations

Austin W.J. and Veletsos A.S. Free vibration ofarches 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.


11/31


Harmonical analysis

Advantages1. The strict analytical solutions are used.

2. The continuity is provided at transition from static to dynamic

3. 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)


12/31

IPS NASU

Dynamical model of pipe bend

as the beam as well as the shell

,02

4

4

WEI

FK

dx

Wd d

A

C

B

D

A

C

B

D

zK

zK

1d

0d

B

O

R

bendpipeforxPfK

pipestraightforK

,,,,

1

B

R

Bt

R2

- flexibility parameter

- parameter of curvature

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

Depends from the frequency !

Physical equation is correctedEI

MK

dx

d

Equation of the transversal vibration with accounting of

increased flexibility:


13/31

IPS NASU

Equation for bend as a shell

r

R

O

B

O1

t

x

y

z

v u

wEquilibrium equations:

0sin1

2

2

0

t

wh

B

N

x

QQ

RR

Nxx

0cos1

2

2

0

t

vh

B

N

x

L

R

QN

Rx

0

sin2

2

0

t

u

hB

Q

x

NL xx

01

x

MM

RQ

x

01

xMM

RQ xxx

HN

HNx

12

HL

HM

HMx

2

1 HM x

Physical equations

Determination of the flexibility

of the pipe bend


14/31

IPS NASU

0

sincos

B

wv

x

u

R

wv

R

12

2

x

w

22

2

2

1

R

ww

R

2

2

x

w

x

w

Rx

v

R

222

deformations

curvatures

Geometrical equations:


of the pipe bend

The simplifications:

semimomentless Vlasovs theory: 0,...,0

vw

x

vR

u,...,

geomtrical characteristics: ,62 BtR 0 B

R

restrictions on the wave length in the axial direction2

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

R

xxx


15/31

IPS NASU


of the pipe bend

Solution for the cylindrical shell

0112

11

,sinsin,,,

6

2

2

22422

2

2

0

n

IV

n

n

VR

hnnnn

ERV

tnxVtxvB

AS

R

R

hnn

nnR

E

m

m

,

112

1

1 2

2

2

222

2

4

22

2

600

800

1000

1200

1400

1600

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

(m, n)

,

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.


16/31

IPS NASU


of the pipe bend

ERBtxVxVRtxv 032 ,sin...3cos,2sin,,,

,2

31, 2 xV

xkxK

The sought for solution:

The resulting equations:

n

IV

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

3,161,1

2,12144,1

22224

,1

22

,1,1

,1

22

,1,1

nnnAnnaBnnaa

nAaBnnaa

nnn

nnn

;133 2,2 nnnAa n

;1 22,4,4h

RRaa nn

;1

11223

1

,4

n

nnna

n

n

;)1()1( 22

22

0

24

hB

RA

;112 42

22 R

h

Ra

;133 2,3 nnnAa n

;1 22,5,5h

RRaa nn

;1

11223

,5

n

nnna

n

n

;3,0

;2,722

nf

nxAkf

n

Eh

RB

2

242 )1(12


17/31

IPS NASU


of the pipe bend

- The coefficient of flexibilityat 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=1

A=3

A=6

A=10A=15

A=30

K

B

Eh

RB 2

242)1(12

BhR2

n

z

IV

n V

EI

FKV

22 ),( Assume:

451

38.28K

ABB

if

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:


18/31

IPS NASU

200

600

1000

1400

1800

2200

2600

3000

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

(m, n)

,

experiment

FEA [Salley and Pan]

our results with

dynamical

K=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.

= 2.07106 ;

= 0.3;= 8000 /3;

R = 0.0806 ;

h = 0.00711 ;

= 0.457 l=0.2

l

l

Rh

B


of the pipe bend


19/31

IPS NASU

P

A B

l

W

thtglPl

M

82

04

42

2

1

EI

lF

tPtP cos

0

-4

-3

-2

-1

0

1

2

3

4

0 50 100 150 200 250 300

M(l/2)

, /c

= 2106; G = 8105;

= 0.3; = 8000 /3;

l= 5 ;R= 0.1 ;h= 0.005 .

1. The graph of bending moment in the

central point of supported-supported beam

srad136

1.25

2. Restoration of the outer force from theknown 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+073.E+07

4.E+07

5.E+07

6.E+07

0 500 1000 1500

, /c

110 P10 W

Abilities of 3D PipeMaster for

vibrodiagnostics


20/31

IPS NASUAbilities of 3D PipeMaster for

vibrodiagnosticsThe problems of vibrodiagnostics

1. The points of application of the outer forces, their directions andfrequencies are unknown.

2. The gauges can measure the displacements of pipe points,

their velocities and accelerations

3. The number of gauges is finite.

The 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


21/31


vibrodiagnostics

yy WF,

A B

ll 20

yQWM ,,

yQWM ,,

tFtFy

sin0

= 2.0689106;= 0.3;

= 7836.6 /3; l= 6.096 ;

l=0.3048;R= 0.05715 ;

t= 0.0188 .

11.66 , 37.65 ,

78.18 .

1 2

3

1. Input data are the results of excitation of beam by harmonical force applied at its

center. The calculated values of transverse forces, bending moment, displacementsin 21 points are recorded. This is so called real case.

2. The system (beam) is loaded by the real displacements in a few (or one) points,

the moments and displacements are calculated.

3. The calculated in 2 results are compared with real data.

The frequency of outer force is given but the point of

its application is unknown. The gauges measure the

displacements


22/31


vibrodiagnostics

A B

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

, de ltaL

W

-600

-400

-200

0

200

400

600

M

W . W "." M . M "."

=21

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 5 10 15 20 25

, de ltaL

W

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

M

W . W "." M . M "."

=8

2 points of measurements


23/31


vibrodiagnostics

=100 =80

=60 -0.00015

-0.0001

-0.00005

0

0.00005

0.0001

0 2 4 6 8 10 12 14 16 18 20

, delta L

W

-600

-400

-200

0

200

400

600

M

W . W "." M . 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

,deltaL

W

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

M

W . W "." M . 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

, deltaL

W

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

M

W . W "." M . M "."


24/31


vibrodiagnostics

A B

l l

=140 =60

2 points ofmeasurements

-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

, del taL

W

-600

-400

-200

0

200

400

600

M

W . W "." M . 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

, de ltaL

W

-600

-400

-200

0

200

400

600

M

W . W "." M . M "."


25/31


vibrodiagnostics

=100 =60

A B

l3 l3 l2 l2

4 points ofmeasurements

-0.00014

-0.00012

-0.0001

-0.00008

-0.00006

-0.00004

-0.00002

0

0.00002

0 5 10 15 20

, deltaL

W

-600

-400

-200

0

200

400

600

M

W . W "." M . 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

, del taL

W

-600

-500

-400

-300

-200

-100

0

100

200

300

400

500

M

W . W "." M . M "."


26/31


vibrodiagnosticsAll measurements in all

points are used Complete coincidence

Conclusions from modeling:

1. To evaluate stresses the most importance have the proximity ofthe points of measurements to the point of the force application.

2. The accuracy grows with the number of the points of

measurement

3. The accuracy nonmonotically decrease with the frequency of

the excitation


27/31

1

,2

form

form

MAXt

MAX

C

ECW


vibrodiagnosticsDetermination of the maximal stresses based

on the measurements of velocities

kt

k

MAX

L

xkAW

EI

mk

ERkL

AEIdx

WdMR

I

M

*sin

)(

,

24

2

2

2

2

2

For simply supported beam:

I

FRE

IE

ERm

WMAXt

MAX

2

For a thin walled pipe:

for a solid circular beam:

For the real complex piping systems:

EWMAXt

MAX

2

EWMAXt

MAX

4

dynamic

susceptibility

coefficient


28/31


vibrodiagnosticsExamples of the piping configuration


29/31


vibrodiagnosticsDetermination of the maximal stresses based onthe measurements of velocities

= 2.06843106;

= 7834 /3; l= 18 ;

R= 0.1 ; t= 0.01 .

J. C. Wachel, Scott J.

Morton, Kenneth E.

Atkins. Piping vibrationanalysis

m

sP10*98.5

7 MAXt

MAX

W

Theoretical value:


30/31


vibrodiagnosticsDetermination of the maximal stresses based onthe measurements of velocities

When the exciting frequency exceeds the first natural frequency the correlation

between the vibrovelocity and maximal stresses is good

= 2.0689106;=7836.6 /3; R= 0.05715 ; t= 0.0188 mFor parameters

Theoretical value

11.66 hertz m

P10*5.69 7 s

WMAXt

MAX

, 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:

1


31/31

IPS NASUConclusion

1. Due to application of dynamical stiffness method the continuitybetween 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 forms

3. In a first time in a literature the notion of dynamic coefficient ofpipe bend flexibility is introduced and analytical expression for it is

obtained. This allowed to perform calculation for the piping

systems with a higher accuracy

4. The option of determination of exciting force in some pointbased on given displacement or velocity in any other point of the

piping allows to efficiently perform the vibrodiagnostic analysis

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10 Ukr Hungary Dynam