2017 5th International Conference on Mechanics and Mechatronics (ICMM 2017)

ISBN: 978-1-60595-541-4

aCorresponding author: wangjun093484@163.com

Design of Single-Point Mooring System with Moored Buoy

Jun WANGa, Yi TANG, Zhu-Yuan YANG and Yu-Mei SHE

School of Mathematics & Computer Science, Yunnan Minzu University, Kunming, Yunnan, China

Abstract. The single-point mooring system is currently under a wide range of applications in marine engineering,

marine observation, marine farming and other engineering fields. So this paper intends to investigate into the single-

point mooring system with moored buoy that is made up of a buoy, four steel pipes, a steel drum, a heavy ball, an

anchor chain and an anchor. The aim here is to design a single-point mooring system mentioned above, which can be

used off shore. Namely, the following parameter values have to be determined: the type and length of chain, the mass

of compact ball, such that do not only both the buoy draught and the moving range, but also the vertical angle of steel

drum reach as small as possible. For the above purpose, firstly, analyse each component of this system by means of

the static force method. Secondly, establish a mathematical model to calculate under the different wind speeds all the

important parameters, including the vertical tilt angle of the steel drum, the horizontal tilt angle of each steel pipe, the

buoy draught, the buoy moving range and the chain shape. Thirdly, on the basis of the above work, a multi-objective

programming model is established to obtain the design schemes in the diverse environmental conditions involving

wind speed, seawater velocity and sea depth. Finally, by simulation it has turned out that these design schemes

obtained above are reasonable and applicable. Therefore, they have definite theoretical significance and practical

values.

1 Introduction

The single-point mooring system with moored buoy is

one of the systems studied earliest by people. And it is

used widely in marine engineering, marine observation,

marine breeding and other fields because of its simple

structure, good direction and low cost. At present, the

study on single-point mooring system with moored buoy

focuses mainly on two aspects: the computational

analysis of and the stability of system. The former mainly

considers tension response of mooring cables and

coupling motion of mooring system. When computational

analysis is performed on the cables, they are generally

recognized as a flexible structure, no withstanding shear

stress, thereby no transmitting torques. The latter studies

mainly the bifurcation and chaotic motion of system and

other dynamic problems. The computational analysis can

be subdivided into the static method and the dynamic

method. The static method [1] neglects the inertial force

of mooring system and involves catenary method, neutral

buoyancy cable method, polygon approximation method,

etc. Smith et al. [2] used the catenary method to do the

static analysis to the mooring system that included two

cables. Therefore, they converted the computational

problem into solving a multi-degree-of-freedom

polynomial equation, which could be solved quickly by

computer. Pan et al. [3] proposed the two-dimensional

static model for a single-point mooring system with

moored buoy, in which the deformation of mooring line

and the change of seawater flow speed were considered.

However, this model was only effective on catenary,

semi-tensioned and tensioned single-point buoy systems.

Wang [4] established the two-dimensional static model

for underwater submerged buoy system, on which the

effect of seawater flow was considered but on which the

effect of the elastic deformation of cable was neglected.

By the static equilibrium, the author calculated some

parameters such as seawater depth, tension, and

inclination tilt angle, horizontal offset and so on, and

developed the calculation software for system design and

deployment. Tang et al. [5] used centralized mass method

for modelling and proposed the calculation method of

mooring tension for deep ocean mooring system. This

method considered gravity, buoyancy, tension, water

current force, submarine support force etc. The results

showed that the greater change of the seabed topography

had a certain influence on the tension of the mooring line.

Lan et al. [6] described in detail the static calculation

steps and methods of submerged buoy system, and

posture analysis. They considered the elongation of cable,

established a three-dimensional static model, and pointed

out that the depth of the submerged buoy system had to

iteratively calculate. Zhang et al. [7] calculated the

tension-span curve for each anchor line based on one

dimensional optimization thought and the catenary

equation method. And the restoring force of mooring

system was calculated by Lagrange interpolation method.

However, this model was only applicable on catenary and

semi-tensioned single-point buoy systems. Wang [8]

197

applied the polygon approximation method based on the

concentrated mass on the static analysis of cable single-

point mooring system and established the static model.

They took into account the elastic stiffness of the cable,

analysed the stress characteristics of typical system

components, including buoy, and established the static

equilibrium equation. For the initial conditions of

uncertainty, the iteration method was used to design the

calculation flow, and the static calculation program was

programmed. The feasibility of this method was verified

by the static calculation of a loose type single point buoy

system. Subsequently, the mooring system was carried

out dynamics analysis and modelling. Wang et al. [9]

studied the deep-ocean single-point mooring system with

moored buoy, performed the static analysis, and

established the static model in which the change of

seawater flow speed with the depth of the water was

taken into account. Recently, some researchers [10-13]

built up the dynamical models for all kinds of mooring

systems to analyse their nonlinear dynamical properties

because of the actual requirements of engineering.

In this paper, an offshore single-point mooring system

with moored buoy is investigated. It is made up of a

buoy, four steel pipes, a steel drum, a heavy ball, a chain

and an anchor. The aim here is to determine the type and

length of chain, and the mass of heavy ball to ensure the

buoy draught, the buoy moving range and the vertical tilt

angle of steel drum are all as small as possible. For this

purpose, the statics analysis is done to the above system

by static method, meantime considering the wind speed,

the seawater speed and the moment of force generated by

four steel pipes and steel drum. Then, to obtain the

optimal design schemes, construct a few mathematical

models to analyse and optimize all kinds of parameters,

including the buoy draught, the buoy moving range and

the vertical tilt angle of steel drum. In the end, it is

verified by simulation that our obtained design schemes

of single-point mooring system with moored buoy are

both reasonable and applicable; thereby they have a

certain practical significance and reference value.

2 The Components of System

2.1 Description of the System Investigated

The seawater depth of the offshore observation network

is between 16m to 20m. Its transmission nodes are

composed of buoy, mooring and underwater acoustic

communication systems. The buoy system is simplified to

a cylinder with mass 1000kg, bottom diameter 2m and

height 2m. The single-point mooring system with moored

buoy consists of four steel pipes (for short “pipe” below),

a steel drum (for short “drum” below), a heavy ball (for

short “ball” below), a welded chain (for short “chain”

below) and an anti-drag anchor (for short “anchor”

below). Each pipe is 1m long, 50mm in diameter, and

10kg in mass. The anchor is 600kg in mass. Usually, a

chain uses common chain link to connect. Its types used

often and the parameters are shown in Table 1. If the

angle between the seabed and the tangential direction of

chain at its end does not exceed 16°, the entire mooring

system will stay there without moving. Otherwise, the

anchor will be dragged such that the nodes are shifted and

hence lost. The underwater acoustic communication

system is installed in a sealed cylindrical drum whose

length and diameter are 1m and 30cm respectively, and

the mass of the whole device is 100kg. The top of drum is

connected with the 4th pipe and its end is connected with

the chain. If the drum is in a vertical state, the underwater

acoustic communication system works best. If it is tilted,

the effect of communication system gets worse. As the

angle between it and the vertical line (referred as vertical

tilt angle) exceeds 5°, the communication system works

very poorly. In order to control the size of vertical tilt

angle, we can hang a heavy ball at the junction of drum

and chain (See Figure 1).

Table 1. Chain types and parameters.

Type (mm)l (kg/m) Type (mm)l (kg/m)

I 78 3.2 IV 150 19.5

II 105 7 V 180 28.12

III 120 12.5

Note 1: The length refers to the one of each chain link.

Steel

Steel Drum

Vertical Line

1

2

3

4

5

Buoy

Heavy Ball Anchor

16 20m m

Sea Level

Seabed

Anchor Chain

3L

1L

2L

4L

5L

Figure 1. The system studied and some symbol descriptions.

2.2 Symbols Description

For the convenience of simplicity, some symbols need to

be introduced. The meaning and its unit of every symbol

(if it existed) are shown in Table 2.

2.3 Fundamental Assumptions

For the sake of simplicity, we gave the following five

fundamental assumptions.

Hypothesis 1: As the mooring system reaches a static

equilibrium, the buoy, 4 pipes, drum, ball, chain and

anchor are all in the same plane.

Hypothesis 2: The volumes of chain can be negligible,

hence both its buoyancy and water current force are zero,

i.e., 6F 6 0A .

Hypothesis 3: 20.625 (N)wF v S approximately.

Here, 2(m )S is the projected area of object on the plane

198

perpendicular to the wind direction, and (m/s)v is the

wind speed.

Table 2. Symbol descriptions.

Symbol Units Meaning

g 2m/s The gravitational acceleration, 9.81g .

H m The water depth.

3kg/m The sea water density, 30251. × 10 .

kg/m The linear density of chain.

Ac

N The current force.

A N The water current force acting on ball.

0 7~A A N The water current force acting on the buoy, the 1st~4th pipe, drum, chain and anchor in sequence.

0 7~m m kg The mass of the buoy, the 1st~4th pipe, drum, chain and anchor in sequence, with 1 2 3 4m m m m .

0 7~G G N The gravity of the buoy, the 1st~4th pipe, drum, chain and anchor in sequence.

1 7~F F N The buoyancy of the 1st~4th pipe, drum, chain and anchor in sequence.

1 5~L L m The lengths of the 1st~4th pipe and drum in sequence, with = , 1 5iL L i .

1 5~D D m The diameters of the 1st~4th pipe and drum in sequence, with =1 2 3 4= =D D D D .

1 5~ The horizontal tilt angles of the 1st~4th pipe and drum in sequence.

0 , ,P Q P The locations of the top 0 0( , )x y , the end 0( , )X -H and arbitrary point ( , )x y of chain respectively.

, , The angles of the sea level and the tangential line at 0, ,P P Q respectively.

0,s s m The lengths of chain from Q to P and to 0P respectively, 0 0s l X .

,h R m The buoy draught and the buoy moving range respectively.

0 0,H D m The height and diameter of buoy respectively.

6,T T N The pulling force of drum acting on chain and on ball respectively.

,G F N The gravity and buoyancy of ball respectively.

7T N The pulling force of chain acting on anchor.

,v u m/s The wind speed and seawater current speed respectively.

( , )x yT N The tension of chain at P .

1T N The pulling force of buoy acting on the first pipe.

iT N The pulling force of the (i-1)-th pipe acting on the i-th pipe, 1 4i .

5T N The pulling force of the 4-th pipe acting on drum.

i

The acute angle formed by the line on which Ti lies and the sea level, 1 4i .

199

The vertical tilt angle of drum, 52

.

0( )F h N The buoyancy of buoy, it is a function with respect to h .

Fw

N The wind force acting on the buoy.

l m The length of chain.

0X m The length of chain “lying down” on the seabed.

m kg The mass of ball.

S 2m The projected area of the object in the normal plane.

( )y f x The shape of chain.

i sin , 1 5T i

i i i

Hypothesis 4: 2374 (N)cA u S approximately. Here,

2(m )S is the projected area of object on the plane

perpendicular to the water current direction, and

(m/s) is the seawater flow speed.

Hypothesis 5: Ocean current is a planar flow field,

and u does not change with the water depth; hence, it is

a constant [9].

Hypothesis 6: The gravity of ball is much larger than

its buoyancy and water current force. So, its buoyancy

and water current force can be negligible, i.e.,

0F A .

Hypothesis 7: The water current force of each

component acts on its centroid.

2.4 The Establishment of Coordinate System

As is shown in Figure 2, a planar coordinate system is

established. According to Hypothesis 1, there exists the

plane where the system reaches the static equilibrium

state. We take it as the coordinate plane.

Take the intersecting line of the coordinate plane and

the sea level as the X-axis whose positive direction is just

the wind direction. And the Y-axis, whose positive

direction is just sticking straight up, passed through the

joint of anchor and chain and is perpendicular to the X-

axis. The intersection of the X-axis and Y-axis is the

origin denoted by O .

H

y

x

R

X

Y

,p x y

0

Anchor

Buoy

0X

Anchor Chain

Sea Level

Seabed

h

Figure 2. The coordinate system.

2.5 Design Process

Step 1: Transmission node uses Type 2 of chain whose

length 22.5ml and the mass of ball m 1200kg .

The anchor is placed on the flat seabed where the water

depth 18mH and its density

3 31.025 10 kg/m . Assuming that 0u , construct a

model to calculate all the parameter values , , , ih R

and determine the shapes ( )f x of chain as 12m/sv

and 24m/sv respectively.

Step 2: According to Hypotheses 1-7, calculate all the

parameter values , , , ih R and determine the shapes

( )f x of chain as 36m/sv . If the anchor is dragged

or the communication system works poorly, then adjust

the mass of ball m, satisfying the conditions that

0 5 & 0 16 .

Step 3: The device is placed in the sea where the

water depth is between 16m and 20m and the maximal

water current speed 1.5m/su and the maximal wind

speed 36m/sv . The design scheme of mooring

system is given according to different wind speed, water

current speed and seawater depth.

Step 4: Simulation. For the designed mooring system,

simulate the static equilibrium system in the following

three cases of 12,24,36m/sv respectively and u

1.5m/s . Namely, determine the reasonability of the

parameter values , , , ih R and the shapes of chain in

the three cases above, as the mooring system is equalized.

Subsequently, verify whether the parameters meet the

demands of objective reality. If not, the design scheme

will be redesigned.

200

3 The Implementation of Step 1

The force analysis of each component is done and then

their force and torque equilibrium equations are achieved

by the above fundamental hypotheses. And so they

constitute a system of equations, through which the

expressions of , , , (1 4)ih R i , and ( )f x can

be deduced. Model 1 is constructed using these

expressions. Then Model 1 is solved and all the

parameter values of , , , ih R and ( )f x is obtained

under different conditions. Now let’s begin with the force

analysis on each component.

3.1 Force Analysis of Each Component

3.1.1 The Force Analysis of Buoy, Pipe and Drum (Including Ball)

By Hypotheses 2 and 6, and Figures 3-5, the force

equilibrium equations of the buoy, 4 pipes (1 4)i

and drum are respectively obtained as follow.

1 1 0 0sin ( )T F h G (1)

1 1cos wT F (2)

1 1sin sini i i i i iT T G F (3)

1 1cos cosi i i iT T (4)

6 5 5 5 5sin sinT T F G G (5)

6 5 5cos cosT T (6)

Further, select respectively as the fulcrum the centroid

C of each pipe and of the drum. Then by Hypothesis 2

and Figures 4-5, their torque equilibrium equations are

also respectively obtained as follow.

1 1sin( ) sin( )i i i i i iT T (7)

5 5 5 6 5 5sin( ) sin( ) cos 0T T G (8)

Wind Direction

Sea Level1

h1T

0F h

0G

wF

Figure 3. Force analysis of buoy.

1

2

1

2

2T

LC

1T1

F

1G

2

Figure 4. Force analysis of the first pipe.

7T

G

3F

6G

5T

5

5

6T

C

7T

Figure 5. Force analysis of drum and ball.

3.1.2 The Global Force Analysis of Chain

The force of chain can be analysed according to the

following two cases:

0 0, 0X ; (2) 0 0, 0X .

By Figure 6, it follows that 0 0s l X . So by

Hypothesis 2, the chain has the following global force

equilibrium equations.

7 6 0sin sin ( )T T g l X

(9)

1 1cos wT F (10)

3.1.3 The Local Force Analysis of Chain

By Figure 7, the local force equilibrium equations are got

as follow.

0

2

01 , 0x

Xs y dx s l X (11)

7 ( , )sin sinx ygs T T

(12)

7 ( , )cos cosx yT T (13)

0 0

, tanx X x Xy H y (14)

Note 2: If 0 0X , then 0 .

201

6T

7T

0gs

0s

0

1 0, 0X

0gs

0s

0

2 0, 0X

0X

0 0 0,P x y

0 0 0,P x y

7T

6T

Figure 6. The global force analysis of chain.

,x yT

7T

gs

0

1 0, 0X

s

0

2 0, 0X

0X

,x yP

gs

,x yT

,x yP

s

7T

Figure 7. The local force analysis of chain.

3.2 Some Basic Relations

By Hypothesis 3 and some basic physical knowledge, the

following basic relations hold.

2

0 00.625 ( )wF D H h v (15)

2

0 0( ) 4F h gD h (16)

24 , 1 5i iF gD L i (17)

6 6

, 1 5;

,

i iG m g i

G m g gl G mg

(18)

3.3 The Establishment of Model 1 and the Results on Solution 3.3.1 The Expression of Relevant Parameters

By Equations 0 0s l X and (1) (18) , the

expressions of , , , (1 4)ih R i and the shape

( )f x of chain are obtained immediately. Let’s discuss

them as follow.

(a) Case 1: 0v . At this time, 0wF . It can be seen

easily that 2 (1 4)i i and 0 . So, it

has the following results :

0 1 5

2 2 2 2

0 1 5 0

00 2

0

00

00

4

4 5 4

4

5

m m m m l

D L l D L D L D HX

D

h H L l X

R X

And the shape ( )f x of chain is composed of the

following two line segments 1C and 2C .

0000

1 2

00

0: :

x Xx t t XC C

y t H t X ly H

(b) Case 2: 0v . Then, 0wF . So it has the

following system of equations.

1 5 0 0 0

0 1 5 6

0 2 2

0 0

4 ( )

4

/ 4 0.625tan

F F F H gX

G G G G Gh H

gD D v

(19)

1 warctan 2 , 1 4i i i F i (20)

5 5 5 5 warctan 2 2F G F (21)

52 (22)

5

0

1

sin i

i

y h L

(23)

wa g F (24)

2

0 01 ( sec ) 1 tans a ay aH (25)

0 0X l s (26)

0 0

2

0 0(tan ) 1 tan1ln

sec tan

x X

s a s a

a

(27)

5

0

1

cos i

i

R L x

(28)

0 00, 0; 0, 0X X (29)

0

0

0 0 0

, 0 ;

seccosh ( )

tansinh ( )

sec,

H x X

a x Xa

ya x X X x x

a

Ha

(30)

Note 3: sini i iT (1 5)i can be obtained by

Equations (1) and (3).

202

3.3.2 The Establishment of Model 1

Now, only consider the case 0v . Combining

Equations (1), (3), (15), (16) and (19) ~ (30), Model 1 is

obtained.

3.3.3 The Solution of Model 1

For Model 1, using the step-search method, solve all the

parameter values , , , (1 4)ih R i as the mooring

system is equalized under different wind speed. Further,

one can ascertain ( )f x . So, one has the following

conclusion.

The mooring system does not shift as 12m/sv and

24m/sv respectively. At this time, all the parameter

values are shown in Table 3, and the shapes ( )f x of

chain shown in Figure 8.

4 The Implementation of Step 2

4.1 Determine the State of Mooring System

4.1.1 The Condition for Mooring System in Static Equilibrium State

The necessary and sufficient condition that the mooring

system reaches a static equilibrium state is 0 0s X l .

4.1.2 Determine the State of Mooring System

Assuming that the mooring system is in a static

equilibrium state as 36m/sv , employ the way used in

Model 1 and finally get 0 0X &0 0.6480s l ,

which does not meet the condition of mooring system in

static equilibrium state.

Table 3. The results of Model 1 ( 12m/sv and 24m/sv ).

(m/s)v 12 24 (m/s)v 12 24

( )h m 0.7348 0.7489 ( )1 89.0236 86.2677

( )R m 14.2983 17.4199 ( )2 89.0178 86.2465

( ) 1.0073 3.8460 ( )3 89.0120 86.2250

( )4 89.0061 86.2033

Figure 8. The shapes ( )f x of chain ( 12 m /s 24 m /sv = ,v = ).

4.2 Determine the Best Mass of Ball

4.2.1 The Establishment of Model 2

According to the above force analysis, getting a model

determining the smallest mass of ball, which is shown

below, called Model 2.

0 0 1 5 6

1 5 0

min 4

tan 4w

G F h G G G G

F F F gX

(31)

0 0

0 5

. . 0 16

1200 ,

(1), (3), (15), (16), (20) (28)

X

s t

m M G mg

Take 2000M as the stop condition for the step-

search algorithm solving Model 2.

4.2.2 The Solution of Model 2

As 36m/sv , the least mass of ball should be

1802.7kg , and at this time, 4.9178 5 and

14.15 16 . The other parameter values are

shown in Table 4.

Table 4. The results of Model 2 ( 36m/sv ).

(m/s)v 36 (m/s)v 36

(m/s)v 36

(m)h 0.9504 ( )1 85.1815 3

( ) 85.1440

(m)R 18.485 ( )2 85.1628 ( )4 85.1250

5 The Implementation of Step 3 — the Design of Mooring System

The design of the mooring system is to determine the

type and length of chain, and the mass of ball, such that

not only do the buoy draught and moving range reach the

minimum, but also the vertical angle of drum do so if the

203

mooring system is in an equilibrium state. Therefore, it is

a multi-objective programming problem.

Based on practical applications, the water current

force has to be taken into account in the system design.

So, it is necessary to make the following modification on

the force equilibrium equations of the buoyant, the drum,

and all the pipes.

5.1. The Modified Equation

5.1.1 The Force Equilibrium Equations of the Buoyant, 4 Pipes and the Drum

There being the water current force, i.e., 0u , so,

according to Hypotheses 2-7, only the horizontal force

equilibrium equations of the buoyant, the 4 pipes and the

drum, i.e., Equations (2), (4) and (6), have to be modified

as follow.

1 1 0 wcosT A F

(32)

1 1cos cos , 1 4i i i i iT T A i (33)

6 5 5 5cos cosT T A (34)

Here,

2

0 0374A D u h (35)

2374 sin , 1 5i i iA D u L i (36)

5.1.2 Modification of the Corresponding Equations

Because of Equations (32), (33) and (34), the other

corresponding equations have to be modified as follow.

1

w 0

arctan ,2

1 4

i ii

i iF A A A

i

(37)

5 5 5

5

w 0 5 5

2arctan

2

F G

F A A A

(38)

w 0 1 5

ga

F A A A

(39)

0 0 1 1 5 5

5

6 0

1

4 4

tan w i

i

G F H G G F F G

F A G gX

(40)

0 1 5 6 1 5

52

0 0 0

1

2 2

0 0

2

0

4 4

tan 0.625

4 0.625 tan

374 tan

i

i

G G G G G F F

gX H D v A

hgD D v

D u

(41)

5.2 The Establishment and Solution of Model 3

5.2.1. The Establishment of Model 3

A multi-objective programming model is built up as

follows.

min ,min &minh R

0 0

0 5. .

0 16

(1), (3), (15), (16), (22) (28), (35) (41)

X

s t

According to Equation (41), for any given ,h is

increasing with respect to G . Similarly, for any given

G , h is increasing with respect to as 36m/sv &

u 1.5m/s . So, convert the multi-objective

programming to the following single-objective

programming.

1 2

min16 5

G RW

M M

0 0

0 5. .

0 16

(1), (3), (15), (16), (22) (28), (35) (41)

X

s t

Here, take 1 6000M & 2 40M .

5.2.2 The Solution of Model 3 and the Design Schemes of Mooring System

Due to 0 5 , it has sin 1i and hence,

2374 , 1 5i iA D u L i . It means that the influence

of i on iA can be negligible (1 5i ).

Set 36m/sv and /s1.5mu in Model 3. If

16,17,18,19,20mH respectively, and 3.2,7,

12.5, 19.5,28.12kg/m , then still use the step-search

method to solve Model 3. Finally, get 25 sub-optimal

solutions of Model 3 under the above 25 kinds of cases.

Select 5 optimal solutions from them as the design

schemes of mooring system, which are shown in Table 5.

204

Table 5. The design schemes of mooring system.

(m)H

16 17 18 19 20

(kg/m)

28.12 28.12 28.12 28.12 28.12

Type V V V V V

(m)l

18.532 19.760 20.964 22.148 23.315

(kg)m

3729.5 3694.9 3661.1 3627.8 3595.0

(m)h

1.6490 1.6490 1.6490 1.6490 1.6490

( )

4.9707 4.9707 4.9707 4.9707 4.9707

(m)R

15.605 16.317 16.988 17.621 18.221

6 The Simulation

In order to verify the theoretical design schemes in Table

5, the simulation is done to observe all the parameter

values as 1.5m/su and 12,24,36m/sv

respectively. Here, it should be pointed out that in the

simulation, the changes of water current force acting on

the 4 pipes and the drum are not taken into account, and

that they are regarded as the maximum values of their

own water current forces instead. Here, the shapes of

chain at different speeds are shown in Figure 9

respectively for the seawater depth 16m in the design

scheme.

Figure 9. The shapes ( )f x of chain.

( 12m/s, 24m/s& 36m/sv v v )

Conclusions

Under the same seawater depths, all the parameters meet

the practical requirements, judging from all the obtained

parameter values, including the shapes of chain, at

different speeds based on the above simulation. So, from

the view of static analysis, the simulation results have

shown that the theoretical design schemes of mooring

system in Table 5 are reasonable and applicable, even if

the hurricane reaches to 36m/s and the seawater current

does 1.5m/s. So they have a certain practical significance

and reference value.

Acknowledgement

This work is financially supported by National Natural

Science Foundation Projects (61462096&11361076),

Yunnan Provincial Department of Education Science

Research Fund Project (2016YJS078), and Yunnan

Minzu University Overseas Master’s Program (3019901).

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