Zhang Kang Wang Lei Leng Wenjun Chen Hongjournal16.magtechjournal.com/.../20200605113328.pdf · The...

CHINESE JOURNAL OF SHIP RESEARCHVOL14NO5OCT 2019To cite this articleZhang K Wang L Leng W J et al Influence of multi-legged pose of the deep-sea crawling-swimming

vehicle on the stability during cruising[J] Chinese Journal of Ship Research 2019 14(5) httpwwwship-researchcomEN Y2019V14I590

DOI1019693jissn1673-3185 01500

Received2019 - 12 - 20Supported by National Key Research and Development Program (2016YFC0301700)Authors Zhang Kang male born in 1994 master Research interest hydrodynamics E-mail heuzhangkang126com

Wang Lei male born in 1979 PhD professor Research interest hydromechanics E-mail ywanglmailustueducnLeng Wenjun male born in 1966 master professor Research interest overall submarine research and design E-maillwj719netChen Hong male born in 1973 PhD professor Research interest control theory and control engineeringE-mail chenhong22001163com

Corresponding authorCHEN Hong

Influence of multi-legged pose ofthe deep-sea crawling-swimming

vehicle on the stability during cruising

Zhang KangWang LeiLeng WenjunChen Hong

Wuhan Second Ship Design and Research InstituteWuhan 430205ChinaAbstract［Objectives］The deep-sea crawling-swimming vehicle（CSV）is a new type of underwater robotof whichthe multi-legged pose will change the distribution of the surrounding flow field and its center of gravity［Methods］Based on the structural characteristics of the CSVie left-right symmetryfront-back approximate symmetry andup-down asymmetrythe maneuvering vertical plane motion equation of CSV was established and the correspondingstability criteria and critical speed were obtained The hybrid mesh was used to compute the hydrodynamic coefficientsof CSVs pure heaving motion and pure pitching motionand the results of computation were compared with theexperimental results According to the stability criterionthe static stability and dynamic stability of CSVs in threeposes including the longitudinal expansion poselateral expansion pose and landing pose were judgedand maininfluence factors of motion stability were analyzed［Results］The results show that CSVs in all three poses are in thestate of static instability and relative dynamic stability The design speed is lower than the critical speedwhich cansatisfy the requirement of linear stability The static stability mainly depends on the position derivative related tovertical force The static stability in lateral expansion pose is the bestwhile that in landing pose is the worst Thedynamic stability mainly depends on the position derivative related to vertical forcethe initial height of stability centerand the structure layout The dynamic stability in lateral expansion pose is the worstwhile that in landing pose is thebest［Conclusions］The hydrodynamic and stability laws of CSVs multi-legged pose can better guide the design ofcontrol system and make it run safelyKey wordsdeep-sea crawling-swimming vehicleautonomous underwater vehicle（AUV）multi-legged poseequation of maneuvering motion in the vertical planehybrid meshstability criteriaCLC number U6611

0 Introduction

The deep-sea crawling-swimming vehicle (CSV)is a new type of non-tethered unmanned underwatervehicle that can cruise in the deep sea and crawl onthe seafloor It also combines the ability of autonomous underwater vehicle (AUV) to be efficient andmaneuverable over a wide range as well as the ability of remotely operated vehicle (ROV) to locate accu

rately as shown in Fig 1 [1-2] The CSV stern has twoducted propellers which can control the cruisingspeed and direction In addition it carries primarysecondary high-power supply which can quicklysink and float through the droppable ballast The total weight of CSV is about 2 000 kg the payload is100 kg the extreme diving depth is 1 000 m thecruising speed is 10 ms the maximum cruisingspeed is 15 ms the maximum crawling speed is

38

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01 ms the maximum working slope is 15deg it canwork normally under the ocean current of 15 ms [1]CSV is mainly composed of the main body propellerand the attached six legs When it cruises in the water different multi-legged poses will change the distribution of the surrounding flow field and its owncenter of gravity affecting its overall force and motion characteristics Therefore mastering the hydrodynamic performance and stability laws at differentposes of CSV is of great significance for its controland safe operation

Previous studies mostly focus on the streamlinedturning vehicles Most analyses of the force of the vehicles use the submarines maneuvering plane motion equations while there is a lack of studies on themaneuvering motion equations of vehicle front-backapproximate symmetry and up-down asymmetry aswell as the cruise motion stability of the attachedmulti-legged vehicle Jun et al [3-5] studied the underwater crabster vehicle CR200 but they did not consider the hydrodynamic force of the underwatermulti-legged vehicle during the cruise Jun et al [3]

proposed an approximate model of the resistance andlift of the crabster legs Kang et al [4] studied the hydrodynamic model of the active control and feedbackcontrol of the crabster vehicle legs and Park et al [5]

obtained the hydrodynamic force and moment basedon the CR200 model test All of them took into account the force exerted on the legs of the underwatermulti-legged vehicle during the crawling process Interms of the stability of vehicles Huang [6] studiedthe influence of the tail and shape on the motion stability of the turning AUV Sun et al [7] studied the motion stability of the measurement deepwater AUV inthe vertical plane All the objects studied are conventional underwater vehicles and the stability criterionof submarines is also used for reference In terms ofthe maneuvering hydrodynamic force of vehicle themulti-function turning underwater vehicle calculated by Zhang [8] the long-endurance underwater vehi

cle studied by Zhang [9] and a new underwater vehicle researched by Xu [10] are also conventional underwater vehicles In summary the current research objects of maneuvering hydrodynamic force and stability are generally streamlined vehicles with concentrated structure but there are few studies on the attached multi-legged vehicle The CSV structure isquite different from the traditional structure of underwater vehicle so the stability criterion cannot be established directly by using the equation of maneuvering plane motion of submarine Meanwhile due tothe complexity of multi-legged structure the mesh isprone to be distorted in the unsteady calculation

This paper will first establish the equation of maneuvering motion in the vertical plane and stabilitycriteria of CSV to verify the effectiveness of the algorithm and hybrid mesh by calculating the ellipsoidmodel Then this paper will use a unified algorithmand approximately consistent mesh division to calculate the maneuvering hydrodynamic force of differentCSV poses and compare the result with the experimental values Finally the advantages and disadvantages of the stability of different CSV poses will beanalyzed according to the criteria1 Maneuvering motion equation

and stability criteria

11 Equation of maneuvering motion inthe vertical plane

The main body of CSV is modeled as an approximate ellipsoid and the six legs are modeled as the attached slender cylinder The structure of CSV isleft-right symmetric front-back approximate symmetric and up-down asymmetric which is quite different from the conventional vehicle According tothis feature the new equation of maneuvering planemotion is established

The fixed coordinate system E - ξηζ and the moving coordinate system G - xyz [11] are adopted whichare the same as the coordinate system set by the submarine research as shown in Fig 2 ( X is the longitudinal force Y is the transverse force Z is thevertical force K is the heeling moment N is theyaw moment M is the pitching moment) The maneuvering equation of CSV in the case of weak maneuvering in the vertical plane is simplified as

igraveiacuteicirc

iuml

iuml

X = muZ = m(w - V )q

M = Iy q（1）

Fig1 Schematic diagram of deep-sea CSV

Zhang K et al Influence of multi-legged pose of the deep-sea crawling-swimming vehicle on the stabilityduring cruising 39


CHINESE JOURNAL OF SHIP RESEARCHVOL14NO5OCT 2019where m is the CSV mass V is the velocity of CSVsteady motion u is the longitudinal accelerationw is the vertical acceleration w is the velocity ofpitch angle q is the acceleration of pitch angle Iy

is the rotational inertia of CSV to the Gy axis

When the model is adopted for the test considering the idea of inertial force and viscous force thehydrodynamic force of CSV is linearized into

igraveiacuteicirc

iuml

iuml

X = X0 + Xuu + XuDuZ = Z0 + Zw w + Zq q + Zw w + Zqq

M = M0 + Mw w + Mqq + Mw w + Mqq（2）

where X0 is the resistance when CSV is in a straightline Du is the change of longitudinal velocity w isthe vertical velocity X0 + XuDu is the resistance ofCSV steady motion Z0 and M0 are respectively zerolift and zero-lift moment Xu Xu are the longitudinal force coefficients induced by longitudinal velocity and acceleration respectively Zw Zw Mw

Mw Zq Zq Mq Mq are the hydrodynamic coefficients with similar meaning

Different from submarines due to left-right symmetry front-back approximate symmetry andup-down asymmetry of CSV there areZq = Mw = M0 = 0 Z0 sup1 0 in Eq (2)

Since CSV is used as weak maneuvering there isu = 0 It is assumed that the combined thrust axis ofthe propeller and the Gx axis are coincided and thenon-hydrodynamic force is taken into account Eq (2)is simplified and substituted into Eq (1) The CSVlinear equation of maneuvering motion in the verticalplane is obtained as follows

igraveiacuteicirc

(m - Zw)w - Zw w - (mV + Zq)q = Z0 + P

(Iy - Mq)q - Mqq - Mw w = Mp + Mθθ（3）

where P is the residual static load Mp is the residual static load moment Mθθ is the righting moment θ is the trim angle

By comparing Eq (3) with the submarines linearequation of maneuvering motion in the vertical plane [11]we can see that the submarine needs to consider thehydrodynamic force generated by the rudder and the

trim moment generated by the propeller Due to thefront-back asymmetry of the submarine the zero liftmoment M0 Zq and Mw are reserved in the linearequation Moreover Zq = Mw = 0 can be realized inthe maneuvering motion equation only when the submarine is considered to be symmetric in its midshipsection12 Stability criterion

When CSV in the vertical plane is subjected tothe transient weak interference only the attack anglechanges and the hydrodynamic force caused by it also changes accordingly If the hydrodynamic forcecaused by slight changes in the attack angle makesthe angle return to its original state CSV is staticallystable Otherwise it is statically unstable Since thechange of attack angle only involves the problem ofangular displacement it is only necessary to analyzethe action trend of the changing hydrodynamic forcemoment M (w) namely the longitudinal relative position of the hydrodynamic center point and the gravity center The situation at this time is the same asthat of the submarine so the same static stability criterion [11] is adopted for the attack angle ie

igrave

iacute

icirc

iumliuml

iumliuml

l α lt 0 Static stability

l α = 0 Neutrally stability

l α gt 0 Static instability

（4）

where the dimensionless hydrodynamic center arm isl α = -M

w Z w Z

w and M w are respectively the verti

cal force coefficient and the pitching moment coefficient induced by the dimensionless vertical velocitywhich is also known as the position derivative andbelongs to the viscous force coefficient

Similar to the situation of submarine it is considered that Z0 has been equalized in the reference motion [11] after aliminating θ with Eq (3) The difference caused by the velocity change is included inthe residual static load P and the residual staticload moment Mp and the response linear equationof the dimensionless attack angle is obtained as follows

A3α + A2α + A1α + A0α = -M θ P （5）

where A0 A1 A2 and A3 are dimensionless coefficients A0 = M

θZ w A1 = M

q Z w - M

θ (m - Z

w) -

M w(m + Z

q) A2 = -M q(m - Z

w) - (I y - M

q)Zw

A3 = (I y - M

q)(m - Z

w) α is the CSV attack angleAccording to Eq (5) the characteristic equation

of the disturbance motion equation can be obtained

K

XG

YZ

Myy

z

N

ζ

η

E ξ

x

Fig2 Fixed coordinate system and moving coordinate system

40


as followsA3λ

3 + A2 λ2 + A1λ + A0 = 0 （6）

where λ is the characteristic root of the characteristic equation

According to the Routh-Hurwitz stability criterion the sufficient and necessary conditions for the dynamic stability of CSV in the vertical plane are as follows

igrave

iacute

icirc

iumliuml

iumliuml

A0 gt 0A2 gt 0A3 gt 0A1 A2 - A0 A3 gt 0

（7）

According to the physical meaning of the maneuvering hydrodynamic coefficient M

θ Z w Z

w

M q and M

q are all negative values Therefore A0 A2 and A3 are all greater than 0 and onlyA1 A2 - A0 A3 gt 0 need to be determined By substituting the specific expressions of coefficients A0 A1

A2 and A3 into Eq (7) we can obtain the dynamicstability criterion of CSV in the vertical plane as follows

igrave

iacute

icirc

iumliuml

iumliuml

l α lt l

q Absolute dynamic stabilityKvd gt 1 Relative dynamic stabilityKvd lt 1 Dynamic instability

（8）

where the dimensionless relative damping force armhas l

q = -M q (m + Z

q) the dynamic stability coefficient in the vertical plane is Kvd = l

q l α + kl

FH l α the

relative force arm of a dimensionless righting moment is l

FH = M θ Z

w the constant coefficient isk =

-Z w(I

y - M q)(m

- Z w)

(m + Z q)[ ]M

q(m - Z w) + Z

w(I y - M

q)+

m - Z w

m + Z q

mprime

is the dimensionless mass M θ =

m gh

V 2 g is the

gravitational acceleration h is the initial metacentric height

When CSV is in a relatively dynamic stable stateit is stable at low speed but not necessarily stable athigh speed From stable to unstable critical velocityVcr according to Eq (8) when Kvd = 1 there is

Vcr =m ghk

Z w(l

q - l α)

（9）If CSV cannot satisfy the absolute dynamic stabili

ty the designed speed should be lower than the critical speed so as to ensure certain the relative dynamic stability

By comparing the stability criterion of CSV with

that of the submarine [11] we can see that the coefficients of the characteristic Eq (6) of the two disturbance motions are different but the final criteria(Eq (8)) are the same This is because when the dynamic stability criterion is established from the characteristic equation of submarines disturbance motion the submarine is assumed to be front-back symmetric which is the same as the structural characteristic of CSV ie Zq = Mw = 0 At this time the characteristic equation of submarines disturbance motion is the same as Eq (6) so the final stability criteria of both are the same13 Maneuvering hydrodynamic coeffi-

cient

When carrying out the pure heaving motion andpure pitching motion simulations of CSV we set themotion law of CSV as shown in Eqs (10) and (11)The vertical force coefficient Z and the pitching moment coefficient M of CSV are calculated The dataare processed according to Eqs (12) and (13) to obtain the relevant maneuvering hydrodynamic coefficients and they are substituted into the criterionEq (8) to determine the stability of the CSV in thevertical plane

igrave

iacute

icirc

iumliuml

iumliuml

ζ = a sinωt

θ = θ = 0w = ζ = aω cosωtw = -aω2 sinωt

（10）

igrave

iacute

icirc

iumliuml

iumliuml

θ = θ0 sinωt

q = θ = θ0ω cosωt

q = -θ0ω2 sinωt

w = w = 0

（11）

igraveiacuteicirc

Z = Za sinωt + Zb cosωt + Z0

M = Ma sinωt + M b cosωt + M0 （12）

igraveiacuteicirc

Z = Zc sinωt + Zd cosωt + Z0

M = Mc sinωt + Md cosωt + M0 （13）

where ζ is the vertical displacement of CSV a ispure heaving motion amplitude of CSV ω is the circular frequency of CSV heaving motion θ is thetrim angle velocity of CSV θ0 is the pure pitchingmotion amplitude of CSV Za = -

aω2 LV 2

Z w in which

L is the main body length of CSV Zb =aωV

Z w

Ma = -aω2 L

V 2M

w M b =aωV

M w Zc = -

θ0ω2 L2

V 2Z

q

Zd =θ0ωL

VZ

q Mc = -θ0ω

2 L2

V 2M

q Md =θ0ωL

VM

q



CHINESE JOURNAL OF SHIP RESEARCHVOL14NO5OCT 2019

2 Calculation model and bound-ary condition

Three typical CSV poses including longitudinal expansion pose (CSV-1) lateral expansion pose(CSV-2) and landing pose (CSV-3) are studied asshown in Fig 3 The model is symmetric about themiddle longitudinal section The main body length isL = 2 700 mm the width is B = 1 260 mm the heightis D = 916 mm The six legs are the same Section 1is modeled as a slender cylinder with a length of L1 =424 mm and a diameter of 85 mm Section 2 is modeled as a slender elliptical cylinder with a length ofL2 = 600 mm The long and short axis of cross sectionellipse are 150 mm and 103 mm respectively Thedistance between the front leg (Leg1) and the middleleg (Leg2) on the same side of the middle longitudinal section is d1 = 727 mm the distance between themiddle leg (Leg2) and the back leg (Leg3) is d2 =783 mm the distance between the two legs on bothsides is 864 mm The computing domain is as follows -5L X 2L -2L Y 2L -2L Z 2L Considering the advantages of limited structuredmeshes and strong adaptability of unstructured meshes the computing domain is divided into the hybridmeshes as shown in Fig 4 (a) The inner domain is asphere with a radius of L which is divided into theunstructured meshes The whole domain minus theinner domain is the outer domain and it is dividedinto the structured meshes During the calculationCSV and the inner domain mesh move together withthe same rule and the mesh deformation only occursat the bottom of the outer domain which can effectively avoid the mesh distortion problem in the caseof unsteady motion The contact is set at the interface between the inner domain and the outer domainand the validation of the hybrid mesh is shown inTable 1 Considering factors such as the calculationaccuracy and time the total number of integral meshes is about 162 million after the mesh independenceanalysis is performed (Table 2)

Table 1 shows the comparison between the calculated value and the test value of direct route resistance of the ellipsoid [12] The calculated value in thetable represents the calculated result of dividing thehybrid mesh of the ellipsoid As can be seen fromthe table the calculation accuracy of the hybridmesh model is very high which can be used to calculate the CSV maneuvering motion

Table 2 shows the calculation results of pure heaving motion of CSV-1 with different meshes at the os

（a）Longitudinal expansion pose（CSV-1）

（b）Lateral expansion pose（CSV-2）

（c）Landing pose（CSV-3）Fig3 Different arrangements of CSVs structure

Leg3 Leg2 Leg1

L1 L2d2 d1

L

D

（a）CSV meshing

（b）Ellipsoid meshingFig4 Computational model and domain meshes

Table 1 Computational and experimental resistance valuesof ellipsoid

Steady motion speedV（mmiddots-1）

051015

Calculatedvalue065316486

Testvalue063301481

Relativeerror

317475104

42


cillation frequency of f = 05 Hz it can be seen thatthe difference in hydrodynamic coefficients with different meshes is small and the selected 162 millionmeshes in this paper have good convergence

Before calculating the user defined function(UDF) is compiled based on the motion rule of theCSV When the UDF program is invoked by Fluentsoftware for calculation the inner domain meshesare set to maintain the same frequency and phase motion as CSV The elastic fairing method is adopted toupdate the dynamic mesh and the mesh distortioncan be avoided effectively when the motion amplitude of the inner domain is smaller than the size ofthe outer domains bottom meshes

This paper selects a separate implicit solver anduses the finite volume method to disperse the computational domain and the governing equation Moreover this paper chooses the standard k - ε turbulence model to seal the governing equation andadopts the PISO algorithm to solve the coupling fieldof velocity and pressure The computational domainuses the velocity inlet boundary condition with aReynolds number of Re = 269 times 106 The boundaryof outlet adopts the free outflow and the boundaryadopts the solid wall condition

When the ellipsoid is calculated and verified thehybrid mesh is divided in the same way as shown inFig 4 (b) The total number of meshes is 320 000and the calculation method and boundary conditionsare set as CSV3 Calculation result and analysis

31 Numerical verification

By simulating the pure heaving motion and thepure pitching motion of the ellipsoid we verified theeffectiveness of the numerical algorithm The inflowvelocity of the pure heaving motion of the ellipsoid isset as 08 ms the amplitude is a = 004 m the oscillation frequencies are f = 04 05 and 06 Hz respectively Five periodic motions are calculated and 400steps are set for each period The amplitude of pure

pitching motion is θ0 = 01 rad and the settings of inflow velocity frequency and period are the same asthat of pure heaving motion

Fig 5 shows the periodic variation curve of thevertical force coefficient in the pure pitching motionof the ellipsoid with time ( Z is the vertical force coefficient) As can be seen from the figure the vertical force coefficient presents a stable periodicchange Furthermore the peak value rises with theincrease in the oscillation frequency and it is consistent with the engineering experience

Table 3 lists the computation and map values ofthe hydrodynamic coefficient of ellipsoid pure heaving motion [11] In the table Z

w and M w are respec

tively the vertical force coefficient and the pitchingmoment coefficient induced by dimensionless vertical acceleration which belong to the inertial force coefficients As can be seen from Table 3 the errors ofthe inertial force coefficients ( Z

w M w ) are relative

ly small Although there is a certain error the viscous force coefficients ( Z

w M w ) are in the same or

der of magnitude Considering that the maneuveringmotion of the ellipsoid belongs to the unsteady motion and the disturbance of the flow field is severeso it is considered that the results obtained in this paper are acceptable The same numerical algorithmand similar meshes are used to calculate the maneu

Table 2 Pure heaving motion hydrodynamic coefficients ofCSV-1 with different meshes

Number ofmeshes73times104

140times104

162times104

233times104

Hydrodynamic coefficientZ

w

-0198-0198-0197-0198

Z w

-0160-0157-0157-0146

M w

0001 3100000 8080000 8040000 823

M w

0144014801480147

Fig5 Periodic variation curves of ellipsoid pure pitchingmotion vertical force coefficients with time

006

004

002

0

-002

-0040 5 10 15

Times

Zprime

f=04 Hzf=05 Hzf=06 Hz

Table 3 Computation and map values of ellipsoid pureheaving motion hydrodynamic coefficients

Hydrodynamiccoefficient

Z w

Z w

M w

M w

Calculatedvalue

-0026 50-0019 760000 150022 10

Mapvalue

-0026 8-0018 6

00018 6

Relativeerror

118624mdash

1886



CHINESE JOURNAL OF SHIP RESEARCHVOL14NO5OCT 2019vering hydrodynamic force of CSV and then the stability of CSV at different poses can be predicted bycombining the same stability criteria32 Calculation result

Pure heaving motion and pure pitching motion ofCSV are studied The motion amplitude is set to bethe same as that of the ellipsoid The inflow velocityis 10 ms and the oscillation frequencies f = 0304 05 and 06 Hz The motion of five periods is calculated with 400 steps in each period

Fig 6 Table 4 and Table 5 show the calculationresults of CSV pure heaving motion and pure pitching motion in three poses Fig 6 shows the curve ofvertical force coefficient of CSV-1 pure heaving motion with time It can be seen from the figure that thevertical force coefficient changes in a stable and periodic manner and the peak value of vertical force coefficient rises with the increase in frequency which isconsistent with the engineering experience

By observing the hydrodynamic coefficients ofCSV in Table 4 and Table 5 we can find that the

magnitude of Mw is small and tends to be zero

while the magnitude of Zq is also small which is

consistent with the front-back approximate symmetric structure characteristics of CSV mentioned inSection 11 In Table 5 Z

q and M q are the vertical

force coefficient and pitching moment coefficient induced by the dimensionless longitudinal trim angleacceleration respectively and they belong to the inertial force coefficient Moreover Z

q and M q are

the vertical force coefficient and pitching moment coefficient induced by the dimensionless longitudinaltrim angle velocity respectively and they belong tothe viscous force coefficient

The hydrodynamic coefficient calculated byCSV-3 pure heaving motion is compared with thetest value and the results are shown in Table 6 Itcan be seen that the relative error of the inertial hydrodynamic coefficient Z

w is small and the magnitudes of the calculated value and the test value ofM

w are all small Furthermore the relative error ofviscous hydrodynamic coefficient Z

w is large andthe relative error of M

w is small Considering thatthe maneuvering motion is unsteady and the flowfield is complex it is thought that the maneuveringhydrodynamic coefficient calculated in this papercan predict the motion stability of CSV

In terms of the static stability of CSV the motionstability of CSV at different poses listed in Table 7 isadopted When l

α of the three crawling poses is allgreater than zero they are all statically unstableMoreover for the value of l

α CSV-3 has the largestvalue and the worst static stability while CSV-2 hasthe smallest value and the best static stability Compared with submarines ( l

α = 020-025) CSV hassome differences in static stability The design andmanipulation of CSV show that the reason why the attack angle is not required to be statically stable isthat the hydrodynamic force evoked by the attack angle will also cause other parameter changes of CSVand whether the attack angle can finally recover issubject to the comprehensive effect of all the forces

Fig6 Periodic variation curves of CSV-1 pure heavingmotion vertical force coefficients with time

0 5 10 15Times

f=03 Hzf=04 Hzf=05 Hzf=06 Hz

04

02

0

-02

-04

Zprime

Table 4 Pure heaving motion hydrodynamic coefficients ofCSV with different poses

Crawling-swimmingpose

CSV-1CSV-2CSV-3


w

-0197-0203-0200

Z w

-0155-0239-0108

M w

0001 140-0000 0680002 020

M w

014701610146

Table 5 Pure pitching motion hydrodynamic coefficientsof CSV with different poses


CSV-1CSV-2CSV-3


q

-0006 33-0007 55-0006 26

Z q

0005 210007 190007 85

M q

-0002 24-0001 86-0001 17

M q

-0054 3-0045 3-0045 4

Table 6 Computation and experiment results of CSV-3pure heaving motion hydrodynamic coefficients


Z w

Z w

M w

M w

Calculatedvalue-02

-01080002 02

0146

Testvalue

-0188-14870010 20157

Relativeerror

657927

-728

44


To analyze the reasons for the differences in thestatic stability of vertical plane at different poses according to the components of l

α (Fig 7) we can seethat the M

w of the deep-sea CSV of the three posesis almost the same and l

α is mainly affected by theposition derivative Z

w The vertical force of CSV-2changes the most dramatically with the vertical velocity while the vertical force of CSV-3 changes mostgently with the vertical velocity Therefore the staticstability of CSV-2 is the best while that of CSV-3 isthe worst

For the dynamic stability of CSV it can be foundfrom Table 7 that the dynamic stability criteria of thethree poses are l

α gt l q gt 0 and Kvd gt 1 and three

poses are all in the relative dynamic stability state inthe vertical plane namely that they are stable in thelow speed zone (not exceeding the critical speed Vcr)CSV-3 has the largest Kvd and the best dynamicstability while CSV-2 has the smallest Kvd and theworst dynamic stability Submarines have high requirements for dynamic stability and are generally inan absolute dynamic stability state [11] CSV is an unmanned underwater vehicle and the length-width ratio is much smaller than that of submarines so thereare differences in the dynamic stability from that ofsubmarines

The main influencing factors of the dynamic stability at different poses are further analyzed As can beseen from Fig 8 l

q l α of the three poses are not sig

nificantly different and the dynamic stability coefficient Kvd is mainly affected by the component

kl FH l

α As can be seen from Fig 9 there is a smalldifference in k among the three poses and the component kl

FH l α is mainly affected by l

α and l FH As

can be seen from Fig 10 l FH is comprehensively af

fected by the position derivative Z w and the righting

moment M θ Since the vertical force on CSV-3

changes most gently with the vertical velocity andthe initial metacentric height is the largest the dynamic stability is the best On the contrary the vertical force on CSV-2 changes most dramatically withthe vertical velocity the initial metacentric height isthe smallest the dynamic stability is the worst

In conclusion the CSVs of the three poses are inthe states of static instability and relative dynamicstability in the vertical plane Different from the stability of the submarine the designed maximum velocity of CSV is about 15 ms which does not reach thecritical speed Vcr It can achieve linear automatic stability and meet the requirements of depth control

Table 7 Motion stability criteria of CSV with different poses


CSV-1CSV-2CSV-3

l α

09470674135

l q

030702540253

l FH

1310703233

k

201203204

Kvd

307249372

Vcr （mmiddots-1）

201184208

Static stabilityStatic instabilityStatic instabilityStatic instability

Dynamic stabilityRelative dynamic stabilityRelative dynamic stabilityRelative dynamic stability

Fig7 Components of vertical plane static stability coefficientsat different poses

Z w M

w l α

CSV-1CSV-2CSV-3

1614121008060402

0-02-04

Fig8 Components of vertical plane dynamic stabilitycoefficient at different poses

l q l

α kl FH l

α Kvd

4035302520151005

0

CSV-1CSV-2CSV-3

Fig9 Components of kl FH l

α at different poses

25

20

15

10

05

0

CSV-1CSV-2CSV-3

l α l

FH k




and safe navigation4 Conclusions

In this paper the stability of CSV in the verticalplane is studied and CSV is modeled to the mainbody with multiple identical appendages For thestructure of CSV with left-right symmetryfront-back approximate symmetry and up-downasymmetry the new equation of maneuvering motionin the vertical plane is established and then the motion stability criterion of CSV is established Thepure heaving motion and pure pitching motion ofCSV are also studied Through the numerical verification and calculation result analysis the followingconclusions are obtained

1) The CSVs of the three poses in the verticalplane are both in the states of static instability andrelative dynamic stability which are different fromthe stability of the submarine The critical velocity isaround 2 ms which can meet the requirements oflinear stability in the design speed (the maximum is15 ms)

2) For the static stability of the three poses the lateral expansion pose (CSV-2) is the best and thelanding pose (CSV-3) is the worst The static stability is mainly affected by the position derivative Z

w For the dynamic stability the lateral expansion pose(CSV-2) is the worst and the landing pose (CSV3) isthe best The dynamic stability is mainly affected bythe position derivative Z

w initial metacentric heightand structure layout

The calculation object in this paper is thenon-streamlined complex structure Due to the com

plexity of the surrounding flow field only the stability of CSV in the vertical plane is studied The nextstep is to carry out numerical and experimental studies on the horizontal stability of CSVReferences［1］ Chen HWang X LWei Wet al Concept and key

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［4］ Kang HYoo S YShim Het al Modeling for the Crabster leg with hydrodynamics force ［C］OCEANS2015-MTSIEEE Washington WashingtonDCUSA

IEEE2015［5］ Park J YShim HJun B Het al Measurement of hy

drodynamic forces and moment acting on CrabsterCR200 using model tests［C］Proceedings of 2017IEEE Underwater Technology BusanSouth KoreaIEEE2017

［6］ Huang M L Research on kinematic stability of autonomous underwater vehicle［D］ TianjinTianjin University2014（in Chinese）

［7］ Sun M YLiu Y HHuang M Let al Design of dynamic stability in vertical plane of autonomous underwatervehicle with measurement missions［J］ MechanicalScience and Technology for Aerospace Engineering201635（9）1402-1407（in Chinese）

［8］ Zhang X P Research on maneuverability and motionsimulation of multifunction vehicle［D］ HarbinHarbin Engineering University2008（in Chinese）

［9］ Zhang H Research on maneuverability and motion simulation of a long-endurance underwater vehicles［D］HarbinHarbin Engineering University2008（in Chinese）

［10］ Xu S F Study on the manoeuvring of an autonomousunderwater vehicle［D］ HarbinHarbin EngineeringUniversity2013（in Chinese）

［11］ Shi S D Submarine maneuverability［M］ BeijingNational Defense Industry Press1995（in Chinese）

［12］ Wang X BHan D F Research on the numerical simulation of drag force on ellipsoid［J］ Journal of Shanghai Scientific Research Institute of Shipping201437（1）14-18（in Chinese）

Fig10 Components of l FH at different poses

0

-005

-010

-015

-020

-025

-030

CSV-1CSV-2CSV-3

Z w M

θ

46


深海爬游机器人多腿位姿对巡游稳定性的影响

张康王磊冷文军陈虹

武汉第二船舶设计研究所湖北武汉 430205

摘要［目的目的］深海爬游机器人是一种新构型潜航器其多腿位姿会改变周围流场与自身重心的分布［方法方法］

针对深海爬游机器人左右对称前后近似对称上下不对称的特点建立深海潜航器的操纵性垂直面运动方程

得到相应的稳定性判据和临界速度采用混合网格计算深海爬游机器人纯升沉和纯俯仰运动的水动力系数

并与试验值进行对比然后结合判据判定纵向展开位姿横向展开位姿和着底位姿这 3种位姿下深海爬游机器

人的静稳定性和动稳定性分析运动稳定性的主要影响因素［结果结果］结果表明深海爬游机器人的 3 种位姿均

处于静不稳定和条件动稳定状态设计航速低于临界速度能满足直线稳定性的要求静稳定性主要受与垂向

力有关的位置导数的影响其中横向展开位姿的最好落地位姿的最差动稳定性的优劣主要受与垂向力有关

的位置导数初稳心高和结构布局的影响其中落底位姿的最好横向展开位姿的最差［结论结论］深海爬游机器人

多腿位姿的水动力和稳定性规律能较好地指导控制设计并使之安全运行

关键词深海爬游机器人自主式水下机器人多腿位姿垂直面操纵性运动方程混合网格稳定性判据

潜艇通气管进气阀阻力特性分析及优化

胡海滨周睿周哲刘义军中国舰船研究设计中心湖北武汉 430064

摘要［目的目的］开展潜艇通气管进气阀结构优化减小进气阻力对潜艇的节能降耗和工作安全具有重要意

义［方法方法］采用计算流体动力学（CFD）方法对进气阀的阻力特性进行模拟分析并通过实验验证计算结果的准

确性同时对阀盘行程和进气角度这 2 项参数进行调节与分析得到进气阀阻力及结构的相关规律［结果结果］

结果显示进气阀压力损失随阀盘行程的增大而逐渐减小但在阀盘行程 h＞244 mm 之后继续增大阀盘行程对

降低压力损失的贡献不大当进气角度θ＞65deg时进气阀压力损失会急剧上升而当进气角度θ＜65deg时角度变

化对进气阀压力损失的影响不大［结论结论］研究表明阀盘行程和进气角度对进气阀阻力特性具有重要影响通

过结构优化可显著降低进气阀的进气阻力所得结果对工程设计具有一定的指导意义

关键词潜艇通气管结构优化阻力特性计算流体力动学

[Continued from page 37]

10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905



01 ms the maximum working slope is 15deg it canwork normally under the ocean current of 15 ms [1]CSV is mainly composed of the main body propellerand the attached six legs When it cruises in the water different multi-legged poses will change the distribution of the surrounding flow field and its owncenter of gravity affecting its overall force and motion characteristics Therefore mastering the hydrodynamic performance and stability laws at differentposes of CSV is of great significance for its controland safe operation

Previous studies mostly focus on the streamlinedturning vehicles Most analyses of the force of the vehicles use the submarines maneuvering plane motion equations while there is a lack of studies on themaneuvering motion equations of vehicle front-backapproximate symmetry and up-down asymmetry aswell as the cruise motion stability of the attachedmulti-legged vehicle Jun et al [3-5] studied the underwater crabster vehicle CR200 but they did not consider the hydrodynamic force of the underwatermulti-legged vehicle during the cruise Jun et al [3]

proposed an approximate model of the resistance andlift of the crabster legs Kang et al [4] studied the hydrodynamic model of the active control and feedbackcontrol of the crabster vehicle legs and Park et al [5]

obtained the hydrodynamic force and moment basedon the CR200 model test All of them took into account the force exerted on the legs of the underwatermulti-legged vehicle during the crawling process Interms of the stability of vehicles Huang [6] studiedthe influence of the tail and shape on the motion stability of the turning AUV Sun et al [7] studied the motion stability of the measurement deepwater AUV inthe vertical plane All the objects studied are conventional underwater vehicles and the stability criterionof submarines is also used for reference In terms ofthe maneuvering hydrodynamic force of vehicle themulti-function turning underwater vehicle calculated by Zhang [8] the long-endurance underwater vehi

cle studied by Zhang [9] and a new underwater vehicle researched by Xu [10] are also conventional underwater vehicles In summary the current research objects of maneuvering hydrodynamic force and stability are generally streamlined vehicles with concentrated structure but there are few studies on the attached multi-legged vehicle The CSV structure isquite different from the traditional structure of underwater vehicle so the stability criterion cannot be established directly by using the equation of maneuvering plane motion of submarine Meanwhile due tothe complexity of multi-legged structure the mesh isprone to be distorted in the unsteady calculation

This paper will first establish the equation of maneuvering motion in the vertical plane and stabilitycriteria of CSV to verify the effectiveness of the algorithm and hybrid mesh by calculating the ellipsoidmodel Then this paper will use a unified algorithmand approximately consistent mesh division to calculate the maneuvering hydrodynamic force of differentCSV poses and compare the result with the experimental values Finally the advantages and disadvantages of the stability of different CSV poses will beanalyzed according to the criteria1 Maneuvering motion equation

and stability criteria

11 Equation of maneuvering motion inthe vertical plane

The main body of CSV is modeled as an approximate ellipsoid and the six legs are modeled as the attached slender cylinder The structure of CSV isleft-right symmetric front-back approximate symmetric and up-down asymmetric which is quite different from the conventional vehicle According tothis feature the new equation of maneuvering planemotion is established

The fixed coordinate system E - ξηζ and the moving coordinate system G - xyz [11] are adopted whichare the same as the coordinate system set by the submarine research as shown in Fig 2 ( X is the longitudinal force Y is the transverse force Z is thevertical force K is the heeling moment N is theyaw moment M is the pitching moment) The maneuvering equation of CSV in the case of weak maneuvering in the vertical plane is simplified as

igraveiacuteicirc

iuml

iuml

X = muZ = m(w - V )q

M = Iy q（1）

Fig1 Schematic diagram of deep-sea CSV






igraveiacuteicirc

iuml

iuml







igraveiacuteicirc

(m - Zw)w - Zw w - (mV + Zq)q = Z0 + P






igrave

iacute

icirc

iumliuml

iumliuml




（4）


w Z w Z




A3α + A2α + A1α + A0α = -M θ P （5）


θZ w A1 = M

q Z w - M

θ (m - Z

w) -

M w(m + Z

q) A2 = -M q(m - Z

w) - (I y - M

q)Zw

A3 = (I y - M

q)(m - Z



K

XG

YZ

Myy

z

N

ζ

η

E ξ

x


40


as followsA3λ

3 + A2 λ2 + A1λ + A0 = 0 （6）



igrave

iacute

icirc

iumliuml

iumliuml


（7）


θ Z w Z

w

M q and M



igrave

iacute

icirc

iumliuml

iumliuml

l α lt l


（8）


q = -M q (m + Z


q l α + kl

FH l α the


FH = M θ Z


-Z w(I

y - M q)(m

- Z w)

(m + Z q)[ ]M

q(m - Z w) + Z

w(I y - M

q)+

m - Z w

m + Z q

mprime


m gh

V 2 g is the



Vcr =m ghk

Z w(l

q - l α)





cient


igrave

iacute

icirc

iumliuml

iumliuml

ζ = a sinωt


（10）

igrave

iacute

icirc

iumliuml

iumliuml

θ = θ0 sinωt


q = -θ0ω2 sinωt

w = w = 0

（11）

igraveiacuteicirc



igraveiacuteicirc




aω2 LV 2

Z w in which


Z w

Ma = -aω2 L

V 2M

w M b =aωV

M w Zc = -

θ0ω2 L2

V 2Z

q

Zd =θ0ωL

VZ

q Mc = -θ0ω

2 L2

V 2M

q Md =θ0ωL

VM

q











Leg3 Leg2 Leg1

L1 L2d2 d1

L

D

（a）CSV meshing




051015


Testvalue063301481

Relativeerror

317475104

42



















140times104

162times104

233times104


w

-0198-0198-0197-0198

Z w

-0160-0157-0157-0146

M w

0001 3100000 8080000 8040000 823

M w

0144014801480147


006

004

002

0

-002

-0040 5 10 15

Times

Zprime




Z w

Z w

M w

M w

Calculatedvalue

-0026 50-0019 760000 150022 10

Mapvalue

-0026 8-0018 6

00018 6

Relativeerror

118624mdash

1886












q and M q are












0 5 10 15Times


04

02

0

-02

-04

Zprime



CSV-1CSV-2CSV-3


w

-0197-0203-0200

Z w

-0155-0239-0108

M w

0001 140-0000 0680002 020

M w

014701610146



CSV-1CSV-2CSV-3


q

-0006 33-0007 55-0006 26

Z q

0005 210007 190007 85

M q

-0002 24-0001 86-0001 17

M q

-0054 3-0045 3-0045 4



Z w

Z w

M w

M w

Calculatedvalue-02

-01080002 02

0146

Testvalue

-0188-14870010 20157

Relativeerror

657927

-728

44













kl FH l



α and l FH As








CSV-1CSV-2CSV-3

l α

09470674135

l q

030702540253

l FH

1310703233

k

201203204

Kvd

307249372


201184208




Z w M

w l α

CSV-1CSV-2CSV-3

1614121008060402

0-02-04


l q l

α kl FH l

α Kvd

4035302520151005

0

CSV-1CSV-2CSV-3



25

20

15

10

05

0

CSV-1CSV-2CSV-3

l α l

FH k


























0

-005

-010

-015

-020

-025

-030

CSV-1CSV-2CSV-3

Z w M

θ

46


























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905






igraveiacuteicirc

iuml

iuml







igraveiacuteicirc

(m - Zw)w - Zw w - (mV + Zq)q = Z0 + P






igrave

iacute

icirc

iumliuml

iumliuml




（4）


w Z w Z




A3α + A2α + A1α + A0α = -M θ P （5）


θZ w A1 = M

q Z w - M

θ (m - Z

w) -

M w(m + Z

q) A2 = -M q(m - Z

w) - (I y - M

q)Zw

A3 = (I y - M

q)(m - Z



K

XG

YZ

Myy

z

N

ζ

η

E ξ

x


40


as followsA3λ

3 + A2 λ2 + A1λ + A0 = 0 （6）



igrave

iacute

icirc

iumliuml

iumliuml


（7）


θ Z w Z

w

M q and M



igrave

iacute

icirc

iumliuml

iumliuml

l α lt l


（8）


q = -M q (m + Z


q l α + kl

FH l α the


FH = M θ Z


-Z w(I

y - M q)(m

- Z w)

(m + Z q)[ ]M

q(m - Z w) + Z

w(I y - M

q)+

m - Z w

m + Z q

mprime


m gh

V 2 g is the



Vcr =m ghk

Z w(l

q - l α)





cient


igrave

iacute

icirc

iumliuml

iumliuml

ζ = a sinωt


（10）

igrave

iacute

icirc

iumliuml

iumliuml

θ = θ0 sinωt


q = -θ0ω2 sinωt

w = w = 0

（11）

igraveiacuteicirc



igraveiacuteicirc




aω2 LV 2

Z w in which


Z w

Ma = -aω2 L

V 2M

w M b =aωV

M w Zc = -

θ0ω2 L2

V 2Z

q

Zd =θ0ωL

VZ

q Mc = -θ0ω

2 L2

V 2M

q Md =θ0ωL

VM

q











Leg3 Leg2 Leg1

L1 L2d2 d1

L

D

（a）CSV meshing




051015


Testvalue063301481

Relativeerror

317475104

42



















140times104

162times104

233times104


w

-0198-0198-0197-0198

Z w

-0160-0157-0157-0146

M w

0001 3100000 8080000 8040000 823

M w

0144014801480147


006

004

002

0

-002

-0040 5 10 15

Times

Zprime




Z w

Z w

M w

M w

Calculatedvalue

-0026 50-0019 760000 150022 10

Mapvalue

-0026 8-0018 6

00018 6

Relativeerror

118624mdash

1886












q and M q are












0 5 10 15Times


04

02

0

-02

-04

Zprime



CSV-1CSV-2CSV-3


w

-0197-0203-0200

Z w

-0155-0239-0108

M w

0001 140-0000 0680002 020

M w

014701610146



CSV-1CSV-2CSV-3


q

-0006 33-0007 55-0006 26

Z q

0005 210007 190007 85

M q

-0002 24-0001 86-0001 17

M q

-0054 3-0045 3-0045 4



Z w

Z w

M w

M w

Calculatedvalue-02

-01080002 02

0146

Testvalue

-0188-14870010 20157

Relativeerror

657927

-728

44













kl FH l



α and l FH As








CSV-1CSV-2CSV-3

l α

09470674135

l q

030702540253

l FH

1310703233

k

201203204

Kvd

307249372


201184208




Z w M

w l α

CSV-1CSV-2CSV-3

1614121008060402

0-02-04


l q l

α kl FH l

α Kvd

4035302520151005

0

CSV-1CSV-2CSV-3



25

20

15

10

05

0

CSV-1CSV-2CSV-3

l α l

FH k


























0

-005

-010

-015

-020

-025

-030

CSV-1CSV-2CSV-3

Z w M

θ

46


























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905



as followsA3λ

3 + A2 λ2 + A1λ + A0 = 0 （6）



igrave

iacute

icirc

iumliuml

iumliuml


（7）


θ Z w Z

w

M q and M



igrave

iacute

icirc

iumliuml

iumliuml

l α lt l


（8）


q = -M q (m + Z


q l α + kl

FH l α the


FH = M θ Z


-Z w(I

y - M q)(m

- Z w)

(m + Z q)[ ]M

q(m - Z w) + Z

w(I y - M

q)+

m - Z w

m + Z q

mprime


m gh

V 2 g is the



Vcr =m ghk

Z w(l

q - l α)





cient


igrave

iacute

icirc

iumliuml

iumliuml

ζ = a sinωt


（10）

igrave

iacute

icirc

iumliuml

iumliuml

θ = θ0 sinωt


q = -θ0ω2 sinωt

w = w = 0

（11）

igraveiacuteicirc



igraveiacuteicirc




aω2 LV 2

Z w in which


Z w

Ma = -aω2 L

V 2M

w M b =aωV

M w Zc = -

θ0ω2 L2

V 2Z

q

Zd =θ0ωL

VZ

q Mc = -θ0ω

2 L2

V 2M

q Md =θ0ωL

VM

q











Leg3 Leg2 Leg1

L1 L2d2 d1

L

D

（a）CSV meshing




051015


Testvalue063301481

Relativeerror

317475104

42



















140times104

162times104

233times104


w

-0198-0198-0197-0198

Z w

-0160-0157-0157-0146

M w

0001 3100000 8080000 8040000 823

M w

0144014801480147


006

004

002

0

-002

-0040 5 10 15

Times

Zprime




Z w

Z w

M w

M w

Calculatedvalue

-0026 50-0019 760000 150022 10

Mapvalue

-0026 8-0018 6

00018 6

Relativeerror

118624mdash

1886












q and M q are












0 5 10 15Times


04

02

0

-02

-04

Zprime



CSV-1CSV-2CSV-3


w

-0197-0203-0200

Z w

-0155-0239-0108

M w

0001 140-0000 0680002 020

M w

014701610146



CSV-1CSV-2CSV-3


q

-0006 33-0007 55-0006 26

Z q

0005 210007 190007 85

M q

-0002 24-0001 86-0001 17

M q

-0054 3-0045 3-0045 4



Z w

Z w

M w

M w

Calculatedvalue-02

-01080002 02

0146

Testvalue

-0188-14870010 20157

Relativeerror

657927

-728

44













kl FH l



α and l FH As








CSV-1CSV-2CSV-3

l α

09470674135

l q

030702540253

l FH

1310703233

k

201203204

Kvd

307249372


201184208




Z w M

w l α

CSV-1CSV-2CSV-3

1614121008060402

0-02-04


l q l

α kl FH l

α Kvd

4035302520151005

0

CSV-1CSV-2CSV-3



25

20

15

10

05

0

CSV-1CSV-2CSV-3

l α l

FH k


























0

-005

-010

-015

-020

-025

-030

CSV-1CSV-2CSV-3

Z w M

θ

46


























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905











Leg3 Leg2 Leg1

L1 L2d2 d1

L

D

（a）CSV meshing




051015


Testvalue063301481

Relativeerror

317475104

42



















140times104

162times104

233times104


w

-0198-0198-0197-0198

Z w

-0160-0157-0157-0146

M w

0001 3100000 8080000 8040000 823

M w

0144014801480147


006

004

002

0

-002

-0040 5 10 15

Times

Zprime




Z w

Z w

M w

M w

Calculatedvalue

-0026 50-0019 760000 150022 10

Mapvalue

-0026 8-0018 6

00018 6

Relativeerror

118624mdash

1886












q and M q are












0 5 10 15Times


04

02

0

-02

-04

Zprime



CSV-1CSV-2CSV-3


w

-0197-0203-0200

Z w

-0155-0239-0108

M w

0001 140-0000 0680002 020

M w

014701610146



CSV-1CSV-2CSV-3


q

-0006 33-0007 55-0006 26

Z q

0005 210007 190007 85

M q

-0002 24-0001 86-0001 17

M q

-0054 3-0045 3-0045 4



Z w

Z w

M w

M w

Calculatedvalue-02

-01080002 02

0146

Testvalue

-0188-14870010 20157

Relativeerror

657927

-728

44













kl FH l



α and l FH As








CSV-1CSV-2CSV-3

l α

09470674135

l q

030702540253

l FH

1310703233

k

201203204

Kvd

307249372


201184208




Z w M

w l α

CSV-1CSV-2CSV-3

1614121008060402

0-02-04


l q l

α kl FH l

α Kvd

4035302520151005

0

CSV-1CSV-2CSV-3



25

20

15

10

05

0

CSV-1CSV-2CSV-3

l α l

FH k


























0

-005

-010

-015

-020

-025

-030

CSV-1CSV-2CSV-3

Z w M

θ

46


























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905




















140times104

162times104

233times104


w

-0198-0198-0197-0198

Z w

-0160-0157-0157-0146

M w

0001 3100000 8080000 8040000 823

M w

0144014801480147


006

004

002

0

-002

-0040 5 10 15

Times

Zprime




Z w

Z w

M w

M w

Calculatedvalue

-0026 50-0019 760000 150022 10

Mapvalue

-0026 8-0018 6

00018 6

Relativeerror

118624mdash

1886












q and M q are












0 5 10 15Times


04

02

0

-02

-04

Zprime



CSV-1CSV-2CSV-3


w

-0197-0203-0200

Z w

-0155-0239-0108

M w

0001 140-0000 0680002 020

M w

014701610146



CSV-1CSV-2CSV-3


q

-0006 33-0007 55-0006 26

Z q

0005 210007 190007 85

M q

-0002 24-0001 86-0001 17

M q

-0054 3-0045 3-0045 4



Z w

Z w

M w

M w

Calculatedvalue-02

-01080002 02

0146

Testvalue

-0188-14870010 20157

Relativeerror

657927

-728

44













kl FH l



α and l FH As








CSV-1CSV-2CSV-3

l α

09470674135

l q

030702540253

l FH

1310703233

k

201203204

Kvd

307249372


201184208




Z w M

w l α

CSV-1CSV-2CSV-3

1614121008060402

0-02-04


l q l

α kl FH l

α Kvd

4035302520151005

0

CSV-1CSV-2CSV-3



25

20

15

10

05

0

CSV-1CSV-2CSV-3

l α l

FH k


























0

-005

-010

-015

-020

-025

-030

CSV-1CSV-2CSV-3

Z w M

θ

46


























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905












q and M q are












0 5 10 15Times


04

02

0

-02

-04

Zprime



CSV-1CSV-2CSV-3


w

-0197-0203-0200

Z w

-0155-0239-0108

M w

0001 140-0000 0680002 020

M w

014701610146



CSV-1CSV-2CSV-3


q

-0006 33-0007 55-0006 26

Z q

0005 210007 190007 85

M q

-0002 24-0001 86-0001 17

M q

-0054 3-0045 3-0045 4



Z w

Z w

M w

M w

Calculatedvalue-02

-01080002 02

0146

Testvalue

-0188-14870010 20157

Relativeerror

657927

-728

44













kl FH l



α and l FH As








CSV-1CSV-2CSV-3

l α

09470674135

l q

030702540253

l FH

1310703233

k

201203204

Kvd

307249372


201184208




Z w M

w l α

CSV-1CSV-2CSV-3

1614121008060402

0-02-04


l q l

α kl FH l

α Kvd

4035302520151005

0

CSV-1CSV-2CSV-3



25

20

15

10

05

0

CSV-1CSV-2CSV-3

l α l

FH k


























0

-005

-010

-015

-020

-025

-030

CSV-1CSV-2CSV-3

Z w M

θ

46


























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905














kl FH l



α and l FH As








CSV-1CSV-2CSV-3

l α

09470674135

l q

030702540253

l FH

1310703233

k

201203204

Kvd

307249372


201184208




Z w M

w l α

CSV-1CSV-2CSV-3

1614121008060402

0-02-04


l q l

α kl FH l

α Kvd

4035302520151005

0

CSV-1CSV-2CSV-3



25

20

15

10

05

0

CSV-1CSV-2CSV-3

l α l

FH k


























0

-005

-010

-015

-020

-025

-030

CSV-1CSV-2CSV-3

Z w M

θ

46


























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905


























0

-005

-010

-015

-020

-025

-030

CSV-1CSV-2CSV-3

Z w M

θ

46


























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905



























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905



Zhang Kang Wang Lei Leng Wenjun Chen Hongjournal16.magtechjournal.com/.../20200605113328.pdf · The...

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