Study on Parametric Resonance of Support Parameters to Flow Pipeline

Abstract

The influence of support parameters on the parametric resonance of fluid conveying pipeline under pulsating flow excitation is studied. Firstly, the mathematical model of fluid conveying pipeline and support system is established, Then the Galerkin method is used for discretization, and the incremental harmonic balance method is used to solve the problem. The effects of support stiffness, support damping, support position, support number and support spacing on the boundary of parametric resonance region and the parametric resonance response are discussed. The results show that under the fixed pulsating amplitude, the pulsating frequency range of system instability becomes smaller with the increase of support stiffness, support damping and the number of support, while the support position and support spacing have less influence on it. The change of support stiffness and support position has little influence on the pulsation frequency that the parametric resonance lasts to the end, but the increase of brace damping makes its value decrease obviously. The research results can provide a theoretical basis for the design and installation of pipeline support system.

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Study on Parametric Resonance of Support Parameters to Flow Pipeline
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MEMA 2020
Journal of Physics: Conference Series 1750 (2021) 012027
IOP Publishing
doi:10.1088/1742-6596/1750/1/012027
1
Study on Parametric Resonance of Support Parameters to Flow
Pipeline
WU Renzhi*1, WANG Ang1, QIN Lei1, WANG Jiapan1
1 School of mechanical and power engineering, Tongji University, Shanghai, 201800,
China
Corresponding author: WU Renzhi
Corresponding author’s workplace e-mail address: wurenzhi@tongji.edu.cn
Abstract. The influence of support parameters on the parametric resonance of fluid conveying
pipeline under pulsating flow excitation is studied. Firstly, the mathematical model of fluid
conveying pipeline and support system is established, Then the Galerkin method is used for
discretization, and the incremental harmonic balance method is used to solve the problem. The
effects of support stiffness, support damping, support position, support number and support
spacing on the boundary of parametric resonance region and the parametric resonance response
are discussed. The results show that under the fixed pulsating amplitude, the pulsating frequency
range of system instability becomes smaller with the increase of support stiffness, support
damping and the number of support, while the support position and support spacing have less
influence on it. The change of support stiffness and support position has little influence on the
pulsation frequency that the parametric resonance lasts to the end, but the increase of brace
damping makes its value decrease obviously. The research results can provide a theoretical basis
for the design and installation of pipeline support system.
1. Introduction
Pipeline is a common device in mechanical hydraulic, chemical, petroleum and natural gas, and large
water conveyance system. When the transport pipeline reaches a certain length, additional pipeline
support is required to satisfy the normal operation of the whole system. The power source of fluid
transmission is generally the pump. Due to the characteristics of the pump itself and the reasons of
processing and manufacturing, the fluid must produce pulsation when flowing in the pipeline[1]. When
the pulsation parameter satisfies a certain relationship, parameter resonance will occur, which will
further affect the working performance of the entire pipeline system[2]. Due to the existence of pipeline
support, the parameter resonance characteristics of the pipeline may change, which will affect the whole
pipeline system. Therefore, it is of great engineering value and theoretical significance to study the
influence of support parameters on parametric resonance.
Zhou Q Z[3] et al. used incremental harmonic balance method to study the influence of support
parameters on Hopf bifurcation point of nonlinear braced cantilever transport pipeline system under the
action of base excitation. Liang F et al.[1,4] used incremental harmonic balance method to study the
parametric resonance of fixed transport tube with two ends under pulsating excitation. Jing S[5] using
nonlinear theory, such as tilted support flow pipeline model is established, support Angle is studied for
flow pipeline support parameters, such as the influence of dynamic response. Bai H H[6] studied the
influence of bearing stiffness change on pipeline stress and other parameters by establishing a fluid-
structure coupling dynamic model of straight pipe with elastic bearing with variable stiffness and finite

MEMA 2020
Journal of Physics: Conference Series 1750 (2021) 012027
IOP Publishing
doi:10.1088/1742-6596/1750/1/012027
2
element simulation. Sheng S W[7] established a pipeline mathematical model by means of transfer matrix
method and analysed the influence of support parameters on the natural frequency and vibration mode
of the pipeline. Jin J D et al.[8,9] studied the parametric vibration characteristics of the hinge points
transport pipeline under the action of pulsating flow and analysed the boundary of the unstable region
under the conditions of different average velocity and mass ratio by establishing the equation of the two-
end hinge points transport pipeline and using the average method. Ariaratnam S T et al[10] studied and
analysed the boundary range of the unstable region under different mean flow rates, boundary conditions
and mass parameters by establishing a transport pipeline model supported by two ends and using the
symbology transformation method and the average method. Namchchivaya N S et al.[11] studied the
subharmonic parameter resonance and combined parameter resonance of the pipeline under the action
of pulsating fluid by establishing nonlinear pipeline model equation and using average method.
Although the above work has comprehensively studied the dynamic characteristics of pipeline
conveying fluid and the influence of support parameters in it, there are few studies on the parametric
resonance of pipeline conveying fluid with support parameters. Therefore, this paper establishes the
dynamic model of pipeline and pipe support, obtains the numerical solution by Galerkin discrete method
and incremental harmonic balance method, and uses numerical examples to study the influence of
support parameters such as support position, support stiffness and support damping on the parametric
resonance of flow conveying pipeline, so as to provide theoretical basis for the design and installation
of pipeline support system.
2. Mathematical Model
2.1. Differential equations of motion for pipe conveying fluid
The vibration of a fixed pipe conveying fluid under the action of pulsating flow is analysed. The pipeline
is placed vertically, and there is an elastic support at 12
= , =
mm
x
xxx
, and its bottom is fixed with a
spacing of
b
21
= mm
x
xx.
Figure1. Support model of pipeline
Assuming that the pipe is an Euler beam model, the influence of shear deformation and gravity is
ignored. The fluid is incompressible and has a steady flow, regardless of the influence of viscosity and
gravity. Only radial vibration of the support is considered for pulsating fluid action. Then the differential
equation of the transport pipeline motion can be written as:





 
2
54 22
2
ff
4 2
400
22
fp f 11 m1 2 2 m2
2
ˆˆ
a12L
2
2 0
LL
yy uEAyEAyyy
EI EI m u T PA m x dx a dx
tLxLxxt
x
x
xt
yy y y
mm mu kyc xx kyc xx
xt t t
t



   

   

 

 
  
  
  
   
  
  
  

 (1)
Where a is the viscoelastic coefficient of the pipeline, E is the elastic modulus of the pipeline material,
I is the moment of inertia of the pipeline section, f
mis the fluid mass per unit length,
p
m is the mass
of the pipeline per unit length, u is the fluid velocity, T is the external tension on the pipeline, PA is the

MEMA 2020
Journal of Physics: Conference Series 1750 (2021) 012027
IOP Publishing
doi:10.1088/1742-6596/1750/1/012027
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static pressure in the pipe,

is the Poisson's ratio, ˆ
A
is the effective cross-sectional area of the pipeline,
L is the length of the pipe, k is the stiffness of the support spring, c is the damping coefficient of the
support, m
x
is the support position,


x

is the

Function, which is defined as a generalized function
describing the distribution density of points.
2.2 Dimensionless
Introduction of dimensionless parameters and the periodic variation of fluid velocity with time is
considered.






11 1
123
2
22 2
2
f 1
1
22
fp fp fp
232
b1m122m2
1m122m2b
1 1
2 2
fp fp
12 ˆ
=; =
fTPA L
mmkL
EI a y x EI t AL
vLu
mm L L mm EI mm EI EI EI
LL
x
cL x k L c L x
LEI LL
EI m m EI m m

  
 

  

 
  

  
 

  
     

，，，， ， ，，，
，，，，，。


 
22 2
01001101
1 cos 2 cos sinvv t v v v t v v t


  

，， (2)
Where 0
v is the dimensionless average velocity, 1

is the dimensionless pulsation frequency,

is
the dimensionless pulsation amplitude.
The dimensionless parameters and equation (2) are substituted into equation (1) to obtain the motion
equation of the pipeline with pulsating flow:



  

 
2
54 2 2
11
2
00 1 0
44 00
22
2
01 10 1 11 m122 m2
22
22cos 2
2cos 1sin =0
vv t v d d
vtv t
   
 
    
  
   


   
    

     
  
  
      
  
  
 (3)
2.3 Discretization of differential equations of motion
The mode function is assumed to be:[4]:
     



   

nn
nnn nn
nn
cosh cos
=cosh cos sin sinh n 1,2, 3...
sinh sin N

     


  
， (4)
Where n

, when n = 1,2, 12
and


are the first two order eigenvalues of the pipeline. According to
the boundary conditions of the fixed support at both ends and referring to the calculation method in
reference 12, it can be obtained that 1

= 4.73, 2

= 7.8532.
The second-order Galerkin expansion is used to discretize equation (4):
     
11 2 2
=qq






， (5)
Write the above formula in matrix form, If

T
12
=


，

T
12
qqq=, then it can be written as:

T
=q


， (6)
By substituting equation (6) into equation (3), both sides are the same left multiplication, and integral
on [0 1], the following results are obtained:







2
01f 0110 1 f
IC2 cos K2 cos sin 0 q+ vB C vC vBE Kqq qq
      


  (7)
where

MEMA 2020
Journal of Physics: Conference Series 1750 (2021) 012027
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doi:10.1088/1742-6596/1750/1/012027
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
   
  
 
4
1m1 1m2
10
1 m1 2 m2 m1 1 m1 2 m1 m2 1 m2 2 m2
4
2m1 2m2
02
22
1T11 1 11 22 1 2
22
0
11 22 1 2 22 2
10 2
I= , C=
01 2
0-
== ,=2 ,
0f
vb
vb
bcq ccqq
BdC
bcc qq c q
 
     
 

 
 


       
 
 

  
   



 

 

，， ,
   
42
10 11
1m1 2m2
42
20 22
22
11
11
TT
11 11 1 11 22 1 2
22
00
22 11 22 1 2 22 2
22
0
2
K= ,
02
1
02
=,E=B+D-C=
01
2
f
g
vc
g
vc
ce
ccq ccqq
CdD d K
ccc qq c q
ec



  







 












 
     


 
 







,,

3. Incremental Harmonic Balance Method
First, a new time variable is introduced
t



By substituting equation (7),



 




2 2
0f00 f
I2cost 2cos sin 0 +CvB Cq KvCt vBE t Kqqqq qq
   


   (8)
Where  is the partial derivative of t.
The first step is the incremental process. Let sum be the solution of the vibration equation, then its
adjacent states are expressed in incremental form as follows:
00 0
= qq q


   ，， (9)
By substituting equation (9) into equation (8) and omitting high-order trace terms,



   














2 2
00 00 f 00 000 f 00
2
00 f 0 0 0 0 00 0 0 00 0
I + 2 cos t 3 K 2 cos sin 3 2
C 2 cos t sin 2 cos t 2 cos sin
qCvB Cq vCt vBEtKqR q
vB Cq vBE tq vB q vC t vBE tq
    

   
    
        

  

(10)
where








2 2
00 0 00 f0 00 0 00 f0
R= - I C 2 cos t K 2 cos sin+qvBCqvCtvBEtKq
   


  (11)
The second step is harmonic balance process.
; qSA qSA
(12)
where
   



 
cs c s
s
12 1 2 s c s
s
T T
j j0 j1 jn j1 j2 jn j j0 j1 jn j1 j2 jn
0 , 1,cos , ..., cos 1 ,sin ,...,sin ,
0
, , ..., , , , ..., , , , ..., , , ,...,
TT
C
SAAAAAACtnttnt
C
Aaa abb b A aa a bb b

 



,,
By substituting equation (12) into equation (10) and (11) and applying Galerkin method, we can
obtain the system of equations with
A
,

,

 as unknown quantity.
Azωε
KAR R R


     (13)
The number of unknowns in equation (13) is two more than that in equation (13). Therefore, one
increment must be selected as the active increment, and the other increment as the reference increment,
and selected as the fixed value. In this way, the unique solution of equation (13) can be solved. In this
paper, the unstable region of parametric resonance and the relationship between vibration amplitude and
pulsation frequency are studied. Therefore,

 is selected as the active increment and

 is the
reference increment, and the initial value 000
, , A


is given. By calculation and error correction, the
solution of equation (13) can be obtained.
4. Influence of Support Parameters on Parametric Resonance Region
When the fluid in the pipe is pulsating, the average velocity, amplitude and frequency of pulsating fluid
will affect the stability of the system [13]. This section analyzes the influence of different support
parameters on the system stability. Referring to reference 4, the parameters of pipeline and fluid are
selected as follows: when the fluid in the pipe has pulsating property, the average velocity, amplitude

MEMA 2020
Journal of Physics: Conference Series 1750 (2021) 012027
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doi:10.1088/1742-6596/1750/1/012027
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and frequency of pulsating fluid will affect the stability of the system. The stability of the support system
is affected by different parameters in this section. According to reference 4, the parameters of pipeline
and fluid are selected as follows: 04, 0.447, 5000, 0v


  
.
The first-order parametric resonance regions with different support stiffness are plotted on the



parameter plane, as shown in figure2. It can be seen from figure2 that increasing the support stiffness
will make the instability region move along the direction of increasing the transverse axis. This is
because the increase of stiffness increases the natural frequency, so the parametric resonance region will
move in the direction of increasing frequency. At the same time, with the increase of support stiffness,
the width of resonance region becomes smaller, that is to say, for the same amplitude, the increase of
support stiffness makes the fluctuation frequency range of instability smaller.
It can be seen from figure3 that the increase of support damping makes the parametric resonance
region move upward along the direction of vertical axis increase, which is due to the increase of support
damping which increases the minimum amplitude of instability. When the amplitude of pulsation is 0.1,
the system will lose stability near the pulsation frequency under condition 1 or condition 2 or condition
3. However, under the condition of condition 4, the system will not lose stability no matter how large
the pulsation frequency is. For the same pulsation amplitude, the increase of support damping reduces
the frequency range of instability.
Figure2. The first order parametric resonance
region with different support stiffness
(1. 111
=0.5, =0, =
m


0,
2. 111
=0.5 , =0 , =
m


100 ,
3. 111
=0.5, =0, =
m


150 ,
4. 111
=0.5, =0, =
m


200 )
Figure3. The first order parametric resonance
region with different support damping
(1. 111
=0.5, =0, =10
m


0,
2. 111
=0.5, =0.1, =
m


100 ,
3. 111
=0.5, =0.2, =
m


100 ,
4. 111
=0.5, =0.5, =
m


100 )
It can be seen from figure4 that changing the value of the support position can make the resonance
region move to the right along the pulsating frequency direction, but it does not reduce the width of the
resonance region, that is, for the same pulsation amplitude, the change of the support position will not
change the fluctuation frequency regions with instability. This is due to the change of the support
position which changes the natural frequency of the system.
It can be seen from figure5 that in the process of changing the number of supports from 0 to 2, the
resonance region moves to the right along the direction of increasing the pulsation frequency. At the
same time, the opening width of the parametric resonance boundary becomes smaller, that is, for the
same amplitude, the increase of the number of supports makes the fluctuation frequency range of
instability smaller. This is due to the increase of the number of supports, which changes the natural
frequency of the system and increases the rigidity and stability of the system.

MEMA 2020
Journal of Physics: Conference Series 1750 (2021) 012027
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doi:10.1088/1742-6596/1750/1/012027
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Figure4. The first order parametric
resonance region with different
(1. 111
=0.1, =0, =10
m


0,
2. 111
=0.2, =0, =
m


100 ,
3. 111
=0.3, =0, =
m


100 ,
4. 111
=0.5, =0, =
m


100 )
Figure5. The first order parametric
resonance region with different support
numbers
(1. Without support, 2. one support,
3. two supports)
It can be seen from figure6 that with the increase of the distance between the two supports, the
resonance region first moves to the right and then to the left along the pulsating frequency direction.
However, for parametric resonance, the V-shaped width of the unstable region does not change, that is,
for the same amplitude, the change of the spacing between the two supports will not change the unstable
pulsation frequency range. This is because the natural frequency of the system first increases and then
decreases with the increase of the distance between the two supports.
Figure6. The first order parametric resonance region
with different spacing between two supports
(1. b1212
=0.2, = =0.1, = =10


 0, 2. 12 12
=0.4, = =0.1, = =10
b


 0, 3. b1212
=0.6, = =0.1, = =10


 0)
5. Influence of support parameters on parametric resonance response of pipeline
The parameters of pipeline and fluid are: 04, 0.447,v

 = 0

5000, . When the support parameters
are 1112 22
=0.5, =0, =0, =0.7, =0, =0
mm
 

, the first-order natural frequency and the second-order
natural frequency of the pipeline corresponding to this parameter value are 12
16.9938, 56.7521
nn



.
The actual amplitude fluctuation frequency response curve in figure7(b) is the pipeline amplitude at
=0.65

. According to figure7(a), when =0.25

, the pulsating frequency values of points a and B
corresponding to the first-order parametric resonance boundary points are AB
30.3, 37.4



. These
two points are at both ends of 1
2 33.9876
n


 . When the pulsating frequency

increases gradually
and reaches A


, the pipeline begins to lose stability and parametric resonance occurs. With the

MEMA 2020
Journal of Physics: Conference Series 1750 (2021) 012027
IOP Publishing
doi:10.1088/1742-6596/1750/1/012027
7
increase of the pulsating frequency

, the actual amplitude

of the pipeline gradually increases, and
this vibration continues to 122

.
Figure7. (a) First order parametric resonance region,
(b) Actual amplitude fluctuation frequency response curve
From the above analysis, given the amplitude of fluid pulsation, when the pulsation frequency
reaches a certain value, the pipeline will lose stability and produce parametric resonance. Therefore, this
section will study the influence of support parameters such as support stiffness, support damping,
support position and support number on parametric resonance response when the fluctuation amplitude
is =0.25

.
The influence of support stiffness on the amplitude A of q1 and q2 modes at first-order parametric
resonance is analyzed by IHB method. figure8 shows the variation of modal amplitudes A1 and A2
when 1111
=0.5, =0, = =10
m


0, 0, 11
=150, 200

. It can be seen from figure8 that with the increase
of support stiffness, the pulsating frequency corresponding to the parametric resonance boundary point
of the first-order 1/2 harmonic resonance due to the instability of the system increases, which is due to
the change of the linear natural frequency of the system due to the existence of the support stiffness. At
the same time, the value of the pulsation frequency which lasts until the end of the resonance increases
correspondingly. Combined with figure8 and figure9, since the modal amplitude A1 is much larger than
A2, q1 is the main component in the actual amplitude (the amplitude at =0.65

). Therefore, in a certain
range, with the increase of stiffness, the change of actual amplitude is similar to that of mode amplitude
A1.
Figure8. Amplitude frequency response
curve of different support stiffness
(1. 111
=0.5, =0, =
m


0,
Figure9. Actual amplitude of pipeline
with different support stiffness
(1. 111
=0.5, =0, =
m


0,
0 20 40 60 80 100 120 140
0
0.1
0.2
0.25
0.3
0.4
0.5
0.6
0 20406080100120140
0
0.02
0.04
0.06
0.08
0.1
0.12
（a）
（b）
BA
q2

MEMA 2020
Journal of Physics: Conference Series 1750 (2021) 012027
IOP Publishing
doi:10.1088/1742-6596/1750/1/012027
8
2. 111
=0.5, =0, =
m


100 ,
3. 111
=0.5, =0, =
m


150 ,
4. 111
=0.5, =0, =
m


200 )
2. 111
=0.5, =0, =
m


100 ,
3. 111
=0.5, =0, =
m


150 ,
4. 111
=0.5, =0, =
m


200 )
It can be seen from figure10 and figure11 that with the increase of support damping, the fluctuation
frequency value corresponding to the boundary point of parametric resonance for the first-order 1/2
harmonic resonance due to system instability changes very little, but the pulsating frequency value of
the parametric resonance lasting to the end decreases obviously, from 126 to 85. Combined with
figures10 and figure11, it can be seen that the variation of the actual amplitude (the amplitude at =0.65

)
is similar to that of the modal amplitude A1.
Figure10. Amplitude frequency response
curve of different support damping
(1. 111
=0.5, =0, =10
m


0,
2. 111
=0.5, =0.1, =
m

100 ,
3. 111
=0.5, =0.2, =
m

100 ,
4. 111
=0.5, =0.5, =
m

100 )
Figure11. Actual amplitude of pipeline
with different support damping
(1. 111
=0.5, =0, =10
m


0,
2. 111
=0.5, =0.1, =
m


100 ,
3. 111
=0.5, =0.2, =
m


100 ,
4.
11 1
=0.5, =0.5, =
m


100 )
Combined with figure12 and figure13, it can be seen that the change of support position makes the
system unstable and the first-order 1/2 harmonic resonance occurs. The fluctuation frequency
corresponding to the boundary point of parametric resonance also changes correspondingly, but the
change of support position has little effect on the fluctuation frequency value of parametric resonance
lasting to the end.
Figure12. Frequency response curves
at different positions
（1. 111
=0.1, =0, =10
m


0,
2. 111
=0.2, =0, =
m


100 ,
3. 111
=0.3, =0, =
m


100 ,
4. 111
=0.5, =0, =
m


100 ）
Figure13. Actual amplitude of pipeline at
different support positions
（1. 111
=0.1, =0, =10
m


0,
2. 11
=0.2, =0, =
m


100 ,
3. 111
=0.3, =0, =
m


100 ,
4. 111
=0.5, =0, =
m


100 ）
0 20406080100120140
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
A1
A2
X: 85
Y: 0.0481
X: 110
Y: 0.06681
X: 118
Y: 0.07249
X: 126
Y: 0.0779
1
2
3
4
0 20 40 60 80 100 120 140
0
0.02
0.04
0.06
0.08
0.1
0.12
3
4
1
2
0 20 40 60 80 100 120 140
0
0.02
0.04
0.06
0.08
0.1
0.12
3
2
1
4

MEMA 2020
Journal of Physics: Conference Series 1750 (2021) 012027
IOP Publishing
doi:10.1088/1742-6596/1750/1/012027
9
6. Conclusion
In this paper, by establishing the mathematical model of pipeline system and support system, the
influence of support parameters on parametric resonance of pipeline system under pulsating flow is
studied by Galerkin method and incremental harmonic balance method. The influence of parameters
such as support stiffness, support damping, support position, support number and support spacing on the
boundary of parametric resonance region is analysed, and the influence of support stiffness, support
damping and support position on parametric resonance response is analysed The results show that: (1)
For a fixed amplitude, the increase of support stiffness and the number of supports makes the fluctuation
frequency range of system instability smaller, while the change of support position and support spacing
does not change the fluctuation frequency range of system instability; the increase of support damping
makes the system parameter resonance boundary move upward along the fluctuation amplitude. (2) With
the increase of support stiffness, the fluctuating frequency corresponding to the boundary point of
parametric resonance will increase. However, the increase of support damping has little effect on the
pulsation frequency corresponding to the boundary point of parametric resonance, and the change of
support position will also increase the critical pulsation frequency of subharmonic resonance of the
system, but the influence is not as great as the change of support stiffness. The results show that the
change of support damping has obvious effect on the pulsation frequency of parametric resonance, while
the change of support stiffness and support position has little effect on it. (3) Through the above analysis,
it is necessary to comprehensively consider the influence of excitation form and support parameters in
the design and installation of pipeline support system to meet the working performance of pipeline
system.
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