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
IPS NASU
Software complex
«3D PipeMaster»

Method of calculation of piping at harmonical vibrations

Modeling of dynamical behavior of pipe bend as the
beam as well as the shell

The abilities of the complex for vibrodiagnostics
Accident of the oil
pipeline
«3D PipeMaster»
Harmonical analysis
IPS NASU
Dynamic stiffness method
with method of initial parameters
y
X1
X0
x
  

  

 X 1    A( , dx)
  

  
 
 
X 0 
 
 
dx
y
X10
stiffness matrix
X11 X20 X21
1
…
2


 i 1 
X 0  




Xn-10 Xn-11 Xn0 Xn1
 
 i
 X 1 ;
 
 
n-1
  
 n 
 X 1    A( )
  
  
n





x
 
 0
 X 0 ; A( ) 
 
 
The sweeping
procedure
n
 A , dx
i
i 1
n i 1

«3D PipeMaster»
Harmonical analysis
IPS NASU
Dynamic stiffness method
 the equations of motion at transversal vibrations
d 4W y
dx
4

F 2
EI z
d z
K
 z
dx
EI z
Wy  0
The inertial term
dK z
 Q y
dx
dW y
dx
 z
 - frequency of vibration
 the equations of the method of initial parameters:
W y  W y0 Y1 k y x  
 z   z Y1 k y x  
0
z
0
ky
K z0
Y2 k y x  
EI z k y
K z  K z0 Y1 k y x  
Q y0
ky
Y2 k y x  
K z0
EI z k
Q y0
2
y
EI z k y2
Y3 k y x  
Q y0
EI z k
3
y
Y4 k y x 
Y3 k y x   k yW y0 Y4 k y x 
Y2 k y x   W y0 k y2 EI z Y3 k y x    z0 k y EI z Y4 k y x 
Qy  Qy0 Y1 k y x   Wy0 k y3 EI z Y2 k y x    z0 k y2 EI z Y3 k y x   K z0 k y Y4 k y x 
k y4

F 2
EI z
«3D PipeMaster»
Harmonical analysis
IPS NASU
The algorithms for branched and curvelinear elements
the conditions in the junctions
2
M  0
Q  0
W 1  W 2  W 3  ...
 1   2   3  ...
1
3
m
equations for pipe bend
X  B( ) X ;
i 1
0
i
i
1
X  C() X ;
n
1
0
0
n
C ( )   Ai  , dxni 1  B(ni 1 ) ;
i 1
3
2
1
5
4
 yb, i1  ye, i cos i  ie sin i ; Wy,bi 1 Wye, i cosi Uie sin i ;
3
2

1
4
The matrix of the turning
element
1
 ib1  ie cos i   ye, i sin i ;
Wz,bi 1 Wz,ei ;
 zb, i1   ze, i ;
Uib1 Uie cosi Wy,ei sin i .
«3D PipeMaster»
Harmonical analysis
IPS NASU
Method of the breaking of displacements for the
determination of the natural frequencies and forms
Xi-10
i-1
Xi-11 Xi0
Xn1
i
i 1
y ,1
W W
i
y,0
y
x
Qyi ,11  Q  0 or
Qyi ,0  Q  0
the criteria of the determination of the natural frequency
Wy ( )  0 
 - natural frequency
The example of the graph Wy ( ) for T –
like frame
i 1
y ,1
Wy  W W
i
y,0
IPS NASU
«3D 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.
«3D PipeMaster»
Harmonical analysis
IPS NASU
Method of the breaking of displacements continuity
The examples of finding the natural frequencies and forms for Tlike frame
1
-1
1
1
0.03
-1
 =148 с-1
The forms given in the handbooks
-1
-1
 =212.4 с-1
-1
 =214.4 с-1
The additional form
of vibration !!!
«3D PipeMaster»
Harmonical analysis
IPS NASU
Method of the breaking of displacements
modeling of curvilinear element
Example: frequencies of the circular ring
n=2
n=3
n=4
n=5
Vibration in the plane of circular ring
theoretical
Our results
167.7051
167.569
474.3416
473.857
909.5086
908.4868
1470.8710
1469.146
Out-of-plane vibration of circular ring
теоретическое
наши результаты
163.6634
468.5213
902.8939
1463.8510
163.36
467.371
900.391
1459.662


2
EI z n 2 n 2  1

,n2
FB04 n 2  1
vibrations in plane


GI y
n2 n2 1

,
4 n2

FB
GI
0
n 2 кр  1
EI y
Out-of-plane
Kang K.J., Bert C.W. and Striz A.G.
Е = 2∙106 МПа; G = 8∙105 МПа; Vibration and buckling analysis of circular
arches using DQM
 = 0.3;  = 8000 кг/м3;
// Computers and Structures. – 1996. –V.60, №1.
В0 = 2 м; R = 0.1 м
– pp. 49-57.
IPS NASU
«3D PipeMaster»
Harmonical analysis
Method of the breaking of displacements
modeling of curvilinear element
Example: 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.
IPS NASU
«3D PipeMaster»
Harmonical analysis
Advantages
1. 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 separated
8. The influence of the subjective factors are excluded (the
breaking out on the elements)
Dynamical model of pipe bend
as the beam as well as the shell
IPS NASU
The curved beam element is strict but pipe bend have the
increased flexibility!
A
B
A
B
R
- parameter of curvature
B
R2
- flexibility parameter

Bt

O
C
Kz
D
Kz
D
C
R
d1
d 0
B
Physical equation is corrected 
d K M

dx
EI
Equation of the transversal vibration with accounting of
increased flexibility:
2
for straight pipe
1
d 4W
d F
K
W  0, K   д
4
dx
EI
K  f  ,  ,  , P, x  for pipe bend
Depends from the frequency !
Determination of the flexibility
of the pipe bend
IPS NASU
Equation for bend as a shell
v
O
1
O
u

r
w
N 1 Q Qx N x sin 
2w



 h 2  0
R R 
x
B0
t

y
t
B
1 N Q L N x cos 
 2v



 h 2  0
R 
R x
B0
t
x
R
z
Physical equations
N  H    
N x  H    
H
L  1    
2
Equilibrium equations:
M  H    
M x  H    
M x
H 1  


2
L N x Qx sin 
 2u


 h 2  0
 x
B0
t
1 M  M x
Q 

0
R 
x
Qx 
1 M x M x

0
R 
x
IPS NASU
Geometrical equations:
 deformations
u v cos   w sin 
  
x
1 v w
 

R  R
B0
 curvatures
1 2w w
   2 2  2
R 
R
Determination of the flexibility
of the pipe bend

2 v 2  2 w


R x R x 
2w
   2
x
2w
   2
x
The simplifications:
 semimomentless Vlasov’s theory:
   0,...,    0
u
v
v
  R ,..., w  

x

2
R
R
 geomtrical characteristics:

 6,    0
B
Bt
2
2

v

v
 restrictions on the wave length in the axial direction

x 2
 2
4
2
1   Q  Q

R   4
 2
3
2
2

 2 N x cos  sin  N x 
  2v  4v 

N


x
 R
  h 2  2  4   0
 2 

2

x  
B0
B0  
t  
 

Determination of the flexibility
of the pipe bend
IPS NASU
Solution for the cylindrical shell
   n n  1 h
E
B  , vx,  , t   V x sin n sin  t ,
 


R n  1  n
121    R


4
0
IV
n
V
2
n




2
 2 2 2
n4 n2 1 h2 


n n 1 
V 0
2
6  n
 ER 2
12
1


R




2
m 
частота, Гц
experiment
FEA [Salley and Pan]
our results
1000
800
600
1,1
1,2
мода колебаний (m, n)
2

,
2

2
Salley L. and Pan J.
A study of the modal
characteristics of curved
pipes // Applied
Acoustics. – 2002. –
V.63. – pp. 189-202.
1400
0,2
2
2
R
A
S
1600
1200
2
m
2
2
2,2
3,2
Determination of the flexibility
of the pipe bend
Решение для гиба
IPS NASU
The sought for solution
:
vx,  , t   RV2 x,  sin 2  V3 x,   cos 3  ...sin  t ,  
K  x,    1 
д
3
2k   x 
V 2  x,  
 B0
ER
The resulting equations:
a1,nVn  a2,nVn2  a3,nVn2  a4,nVn1  a5,nVn1  aVnIV  f n




a1,n  a1,n  n 2 n 2  1 B,

2
2
a1,n  a1,n  n n  1 B,
a1,n  144  12 A,


R
a4 , n  a4 , n  1   R ;
h
3
2
n 1 n  n  n
a4,n  12   1 
;
n 1
2
R 4 (1   2 )
A
 (1   2 )2 ;
2 2
B0 h


R2 4
a  12 1  
R ;
h2
2

2


a1,n  n 4 n 2  1  6 A  n 2 n 2  1 ,
a2,n  3 A  n 2 3  n1  n;
2

n2
n3
a3,n  3 A  n 2 3  n1  n;


R
;
h
3
2
n n n n
 12   1 
;
n 1
a5,n  a5,n  1   2 R 2
a5,n
 f 2  72 Ak x , n  2;

n  3;
 f n  0,
12(1   2 ) R 4 2
B
h2 E
Determination of the flexibility
of the pipe bend
IPS NASU
Assume: V
if
IV
n

FK д ( ,  2 ) 2
EI z
28.8  3 A
B  B 
1  K д 45
Vn
30
25
20
then we obtain :




A=1
A=3
A=6
A=10
A=15
A=30
15
a1,n  a1,n  n 2 n 2  1 B,

n2
 a1,n  144  12 A,


a1,n  a1,n ,



2
4
2
2
2
a1,n  n n  1  6 A  n n  1 , n  3

Kд
10
5

0
10
20
A  (1   )
2
Results:
2
- The coefficient of flexibility at harmonical vibrations
K
 72  6 A  11B  
B
0
72  6 A  11B 2  4B72  60 A  10B 
2B
30
40
50
12(1   2 ) R 4 2
B
h2 E
60
R2

Bh
Determination of the flexibility
of the pipe bend
IPS NASU
L. Salley and J. Pan. A study of the modal characteristics of curved pipes
// Applied Acoustics. – 2002. – V.63. – pp. 189-202.
experiment
3000
FEA [Salley and Pan]
2600
B
our results with
dynamical К
K=1
l
l
2200
the Saint-Venant
(static) solution
частота, Гц
h
R
1800
1400
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
1000
600
200
R,2s
R,2a
1,2s
1,2a
3,2s
мода колебаний (m, n)
3,2a
1,1a
Abilities of «3D PipeMaster» for
vibrodiagnostics
IPS NASU
W
W0  1 P
A
Pt   P0 cos t
P0  1
Е = 2∙106 МПа; G = 8∙105 МПа;
= 0.3;  = 8000 кг/м3;
l = 5 м; R = 0.1 м; h = 0.005 м.
х
B
1
l
1. The graph of bending moment in the
central point of supported-supported beam
1 4 F 2 l 4

2
EI
l  Pl
M    0 tg  th 
 2  8
  136 rad s
3
2
частота, рад/c
-3
-4
4.E+07
3.E+07
1.E+07
0
-2
P0, H
2.E+07
1
-1
6.E+07
5.E+07
M(l /2)
4
2. Restoration of the outer force from the
known displacements in arbitrary point
0
50
100
1.25
150
200
250
300
0.E+00
-1.E+07 0
-2.E+07
-3.E+07
-4.E+07
500
1000
1500
частота, рад/c
IPS NASU
Abilities of «3D PipeMaster» for
vibrodiagnostics
The problems of vibrodiagnostics
1. 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 accelerations
3. The number of gauges is finite.
The functions of the calculation software
1. 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
Abilities of «3D PipeMaster» for
vibrodiagnostics
IPS NASU
Fy ,Wy
х
A
B
M ,W , Q y
M ,W , Q y
l  20l
Fy t   F0 sin t
Е = 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.
1  11.66 Гц, 2  37.65 Гц,
3  78.18 Гц.
The frequency of outer force is given but the point of
its application is unknown. The gauges measure the
displacements
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, displacements
in 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».
Abilities of «3D PipeMaster» for
vibrodiagnostics
IPS NASU
х
A
2 points of measurements
B
3l 3l
0.003
2500
0.0001
2000
0.0025
0
0
1500
2
4
6
8
10
12
14
16
18
20 400
-0.0001
1000
0.002
-0.0002
500
0.001
0
-500
-0.0004
-1000
-0.0005
-1500
0.0005
-0.0003
W
0
200
-200
-0.0006
-400
-2000
0
-2500
0
5
10
15
20
25
-0.0007
-0.0008
-600
Длина, deltaL
W восст.
W "реал."
 =8 Гц
M восст.
Длина, deltaL
M "реал."
W восст.
W "реал."
M восст.
 =21 Гц
M "реал."
M
0.0015
M
W
600
Abilities of «3D PipeMaster» for
vibrodiagnostics
IPS NASU
0.0001
600
400
0.00005
200
0
W
2
4
6
8
10
12
14
16
18
20
0
M
0
-0.00005
-200
-0.0001
-400
-0.00015
-600
 =60 Гц
Длина, deltaL
W восст.
W "реал."
M восст.
M "реал."
0.0008
4000
0.0005
4000
0.0006
3000
0.0004
3000
0.0004
2000
0.0002
0.0003
2000
0.0002
1000
1000
2
4
6
8
10
12
14
16
18
0
W
0
0
0
20
0
-0.0002
-1000
-0.0004
-2000
-0.0006
-3000
-0.0003
-4000
-0.0004
-0.0008
2
4
W "реал."
 =80 Гц
M восст.
8
10
12
14
16
18
-3000
-4000
Длина, deltaL
M "реал."
20 -1000
-2000
-0.0002
Длина,deltaL
W восст.
6
-0.0001
W восст.
W "реал."
M восст.
 =100 Гц
M "реал."
M
0
M
W
0.0001
Abilities of «3D PipeMaster» for
vibrodiagnostics
IPS NASU
2 points of
measurements
х
A
B
l
l
0.00002
0
0
2
4
6
8
10
12
14
16
18
600
0.00006
600
20 400
0.00004
400
200
0.00002
200
0
0
0
0
-0.00008
2
4
6
8
10
12
14
16
18
20
-200
-0.00002
-200
-400
-0.00004
-400
-600
-0.00006
-0.0001
-0.00012
-0.00014
-600
Длина, deltaL
Длина, deltaL
W восст.
W "реал."
M восст.
 =60 Гц
M "реал."
W восст.
W "реал."
M восст.
 =140 Гц
M "реал."
M
-0.00006
M
W
-0.00004
W
-0.00002
Abilities of «3D PipeMaster» for
vibrodiagnostics
IPS NASU
4 points of
measurements
х
A
B
2l 3l 3l
2l
0.00002
600
0
0
5
10
15
20
400
-0.00002
200
-0.00008
-200
-0.0001
-400
-0.00012
-0.00014
-600
400
0.00004
300
0.00002
200
-0.00002
100
0
2
4
W "реал."
M восст.
 =60 Гц
8
10
12
14
16
18
0
-100
-0.00006
-200
-0.00008
-300
-0.0001
-400
-0.00012
-500
-0.00014
-600
Длина, deltaL
M "реал."
20
-0.00004
Длина, deltaL
W восст.
6
W
0
0.00006
M
-0.00006
500
0
M
W
-0.00004
0.00008
W восст.
W "реал."
M восст.
 =100 Гц
M "реал."
IPS NASU
Abilities of «3D PipeMaster» for
vibrodiagnostics
All measurements in all
points are used
Complete coincidence
Conclusions from modeling:
1. To evaluate stresses the most importance have the proximity of
the 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
IPS NASU
Abilities of «3D PipeMaster» for
vibrodiagnostics
Determination of the maximal stresses based
on the measurements of velocities
For simply supported beam:
M
d 2W
2 2
  R, M  2 EI   MAX  A 2 k ER
I
dx
L
mk2
4
(k 
)
EI
x
Wt  A sin k
* k
L
 MAX
 2 E
For a thin walled pipe:

Wt MAX
for a solid circular beam:
 MAX
 4 E

Wt MAX
For the real complex piping systems:
 MAX
m ER



IE
Wt MAX
FR 2
 E
I
dynamic
susceptibility
coefficient
 MAX
 C form 2 E ,
Wt MAX
C form  1
IPS NASU
Abilities of «3D PipeMaster» for
vibrodiagnostics
Examples of the piping configuration
IPS NASU
Abilities of «3D PipeMaster» for
vibrodiagnostics
Determination 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.
Theoretical value:
 MAX
Pа  s
 5.98 *107
Wt MAX
m
J. C. Wachel, Scott J.
Morton, Kenneth E.
Atkins. Piping vibration
analysis
IPS NASU
Abilities of «3D PipeMaster» for
vibrodiagnostics
Determination of the maximal stresses based on
the measurements of velocities
For parameters
Theoretical value
1  11.66 hertz
ω, Гц
 MAX
Wt MAX
obtained value
Е = 2.0689∙106 МПа; ρ=7836.6 кг/м3; R = 0.05715 м; t = 0.0188 m
 MAX
Pа  s
 5.69 *107
Wt MAX
m
The results of calculation:
2
8
21
40
60
80
3.08E+08
7.78E+07
2.87E+07
5.71E+07
5.12E+07
5.55E+07
5.4
1.4
0.51
1
0.9
0.98
When the exciting frequency exceeds the first natural frequency the correlation
between the vibrovelocity and maximal stresses is good
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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 forms
3. 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 accuracy
4. 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