10 ukr hungary dynam
TRANSCRIPT
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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
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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
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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
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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
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IPS NASU3D PipeMaster
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
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IPS NASU3D PipeMaster
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
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IPS NASU3D PipeMaster
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.
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IPS NASU3D PipeMaster
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
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IPS NASU3D PipeMaster
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
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IPS NASU3D PipeMaster
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.
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IPS NASU3D PipeMaster
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)
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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:
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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
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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:
Determination of the flexibility
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
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IPS NASU
Determination of the flexibility
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.
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IPS NASU
Determination of the flexibility
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
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IPS NASU
Determination of the flexibility
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:
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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
Determination of the flexibility
of the pipe bend
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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
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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
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IPS NASUAbilities of 3D PipeMaster for
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
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IPS NASUAbilities of 3D PipeMaster for
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
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IPS NASUAbilities of 3D PipeMaster for
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 "."
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IPS NASUAbilities of 3D PipeMaster for
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 "."
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IPS NASUAbilities of 3D PipeMaster for
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 "."
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IPS NASUAbilities of 3D PipeMaster for
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
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1
,2
form
form
MAXt
MAX
C
ECW
IPS NASUAbilities of 3D PipeMaster for
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
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IPS NASUAbilities of 3D PipeMaster for
vibrodiagnosticsExamples of the piping configuration
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IPS NASUAbilities of 3D PipeMaster for
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:
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IPS NASUAbilities of 3D PipeMaster for
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
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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