Research ArticleResearch on the Numerical Simulation of the NonlinearDynamics of a Supercavitating Vehicle
Tianhong Xiong1 Xianyi Li2 Yipin Lv1 and Wenjun Yi1
1National Key Laboratory of Transient Physics Nanjing University of Science and Technology Nanjing 210094 China2College of Mathematical Science Yangzhou University Yangzhou 225002 China
Correspondence should be addressed to Xianyi Li mathxyliyzueducn
Received 27 April 2016 Accepted 21 July 2016
Academic Editor Dumitru I Caruntu
Copyright copy 2016 Tianhong Xiong et alThis is an open access article distributed under theCreativeCommonsAttributionLicensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Little is known about the movement characteristics of the supercavitating vehicle navigating underwater In this paper based ona four-dimensional dynamical system of this vehicle its complicated dynamical behaviors were analyzed in detail by numericalsimulation according to the phase trajectory diagram the bifurcation diagram and the Lyapunov exponential spectrum Theinfluence of control parameters (such as various cavitation numbers and fin deflection angles) on the movement characteristicsof the supercavitating vehicle was mainly studied When the system parameters vary various complicated physical phenomenasuch as Hopf bifurcation periodic bifurcation or chaos can be observed Most importantly it was found that the parameter rangeof the vehicle in a stable movement state can be effectively determined by a two-dimensional bifurcation diagram and that thebehavior of the vehicle in the supercavity can be controlled by selecting appropriate control parameters to ensure stable navigation
1 Introduction
Liquid vaporization of a liquid occurs at any point when thepressure at that point is reduced below a critical value Inthe initial stage the above phenomenon is microscopic Astime progressesmacroscopically small bubbles arise Furtherthese macroscopic bubbles join to form larger cavum ofsteam and gas in the interface of the liquid or liquid andsolid known as cavities [1] The emergence developmentand crumble processes of the cavity are called cavitationphenomena Supercavitation is a state in which the cavityappears on the whole surface of the object and in the liquidnear the end In this state the formed cavity is like a big steambag exceeding the end of the object or loading the entireobject inside hence the name is supercavity [1ndash3]
A dimensionless cavitation number 120590 that reflects thecavity is introduced to investigate the characteristic of thesupercavity in a general case The cavitation number 120590 isdefined as 120590 = (119875
infinminus 119875119888)05120588119881
2 where 119875infin
is the ambientpressure 119875
119888is the cavity pressure 120588 is the water density and
119881 is the vehicle velocity [1] Once the supercavity becomes
stablemost of the vehiclersquos surface is surrounded by gases andthe resistance of the vehicle decreases sharply This increasesthe navigation velocity and the distance travelled by thevehicle [2ndash4] However when a supercavitation vehicle isnavigating at high speed under the water most of the vehicleis surrounded by the cavity the wet area will be significantlyreducedwhich results in the loss ofmost of the buoyancyTheparts of the vehicle that are in contact with water are mainlya cavitator in the front and fins at the rear of the vehicleWhen coming into contact with the cavity wall the fin willproduce complex nonlinear planing force whichwill increasethe frictional resistance of the vehicle causing vibrations andimpact to the vehicle [5ndash10] Hence the key to ensuringstable underwater navigation of the vehicle lies in effectivelycontrolling the behavior of the supercavitation vehicle andreducing the impact of the collision between the vehicle andthe cavity wall
Based on a four-dimensional dynamical system of asupercavitating vehicle its complicated physical phenomenawere studied bymeans ofmultiple dynamical analysis aimingat the complex nonlinear planing force generated by the
Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 8268071 10 pageshttpdxdoiorg10115520168268071
2 Shock and Vibration
contact between the fins and the cavity walls The mostimportant finding of this study was that the region andparameter range of the vehicle in a stable movement stateare determined by the two-dimensional bifurcation diagramThe movement characteristics of the supercavitation vehicleunder different control parameters were also discussed indetail
To the best of our knowledge it is very difficult to find anyrelated work in this paper up till now
2 Dynamic Modeling ofthe Supercavitating Vehicle
21 Force Analysis When the underwater vehicle is navi-gating at high speed in the supercavitating state most ofthe vehicle will be enveloped by the cavity and only asmall part of the surface will have a contact with the waterWhile the cavitator at the front has a direct contact withwater the cavitator can rotate by a certain angle Differenthydrodynamic power can be provided for the vehicle withthe change of the angle and a planing force can be producedwhen the fin contacts with the cavity wall Four fins weresymmetrically arranged in the rear part of the vehicle Apart of the fin penetrates the cavity wall to have a directcontact with the water thus providing the required forceand momentum to stabilize and control the vehicle with thecavitator In this case the control surface is composed of thecavitator and the four fins [5] The deflection angles of thecavitator and the fins are usually selected as the feedbackcontrol inputs to ensure the stable underwater movementof the vehicle The shape and the force diagram of thesupercavitating vehicle are presented in Figure 1
The forces acting on the vehicle in its own coordinatesystem are indicated in Figure 1 The main forces include thelift force on the cavitator 119865cavitator the lift force on the fin119865fins the gravity at the centroid of the vehicle 119865gravity andthe planing force generated by the interaction between the finand the cavity wall 119865planing the last of which is a complicatednonlinear planing force that consequently causes vibrationand impact to the vehicleThe expression of the planing forceis as follows [11]
119865planing = minus1198812
[1 minus (
1198771015840
ℎ1015840+ 1198771015840)
2
](
1 + ℎ1015840
1 + 2ℎ1015840)120572 (1)
where 119881 is the velocity of the vehicle and 1198771015840
= (119877119888minus
119877)119877 where 119877119888and 119877 are the radius of the cavity and the
vehicle respectively The immersion depth of the aft of thesupercavitating vehicle ℎ1015840 is given as follows [11]
+ (119908 minus 1199081199050) tanh [119896 (119908 minus 119908
1199050)]
(3)
Fin w
q
V
Ffin
FplaningFgravity Cavitator
FCavitator
Figure 1 Shape and force diagram of the supercavitating vehicle
0020 0024 0028 0032 0036minus60
0
20
40
minus20
minus40
120590
k
Figure 2 Dynamic behavior distribution diagram of the system
where the positive value of 119908 at the transition point is1199081199050
= (119877119888minus 119877)119881119871 and 119896 is a constant used to control the
approximated error which is generally set to 300The geometrical angle between the vehicle centerline
and the cavity centerline is the immersion angle 120572 of thesupercavitating vehicle expressed as [11]
120572 =
119908
119881
minus tanh (119896119908)119888
119881
(4)
where 119877119888is the cavity radius and
119888is the shrinkage ratio at a
distance 119871 from the cavitator
22 Dynamic Modeling Through the interactive relationshipbetween the vehicle and the cavity obtained in the previoussection the model can be established based on the forceequations [12] According to the coordinate system presentedin [13] the origin is the center of the disk cavitator at thefront of the supercavitating vehicle 119883-axis aligns with thesymmetry axis of the vehicle and points forward 119885-axis isperpendicular to119883-axis facing vertically downward and119908 isthe velocity in119885-axis direction119881 represents the longitudinalvelocity of the cavitator at the vehicle front and 120579 119902 and 119911
are the pitching angle pitching rate and depth of the vehiclerespectively
Shock and Vibration 3
120590
w
40
30
20
10
000360032002800240020
Figure 3 Bifurcation diagram varying with 120590 when 119896 = 1
13
12
11
10
09
0032500323003210031900317
120590
w
Figure 4 Bifurcation diagram for 00315 lt 120590 lt 00325
According to the theory of rigid body dynamics thefollowing relationship can be derived relating the abovevariables [13]
(
120579
) = (
0 1 minus119881 0
0 11988622
0 11988624
0 0 0 1
0 11988642
0 11988644
)(
119911
119908
120579
119902
)
+(
0 0
11988721
11988722
0 0
11988741
11988742
)(
120575119890
120575119888
) +(
0
1198882
0
0
)
+(
0
1198892
0
1198894
)119865planing
(5)
where
11988622=
119862119881119879
119898
(
minus1 minus 119899
119871
) 119878 +
17
36
119899119871
11988624= 119881119879119878 (119862
minus119899
119898
+
7
9
) minus 119881119879(119862
minus119899
119898
+
17
36
)
17
36
1198712
11988642=
119862119881119879
119898
(
17
36
minus
11119899
36
)
11988644=
minus11119862119881119879119899119871
36119898
11988721=
1198621198812
119879119899
119898
(
minus119878
119871
+
17119871
36
)
11988722=
minus1198621198812
119879119878
119898119871
11988741=
minus111198621198812
119879119899
36119898
11988742=
171198621198812
119879
36119898
1198882= 119892
4 Shock and Vibration
30
20
0
minus10
minus20200minus20minus40minus60minus80
k
w
10
Figure 5 Bifurcation diagram varying with 119896 when 120590 = 00315
15
10
05
0
minus05
k
w
minus50minus55minus60minus65minus70minus75minus80
Figure 6 Bifurcation diagram for minus80 lt 119896 lt minus50
minus0001 0 0001006
007
008
009
010
Equilibrium point
q (rads )
w(m
middotsminus1)
(a)
0 5 10 15
minus100
minus80
minus60
minus40
minus20
0
20
Lyap
unov
expo
nent
spec
trum
t (s)
(b)
Figure 7 (a) Phase trajectory diagram in 119908 minus 120579 plane and (b) Lyapunov exponent spectrum
1198892=
119879
119898
(
minus17119871
36
+
119878
119871
)
1198894=
11119879
36119898
119878 =
11
60
1198772
+
1331198712
405
119879 =
1
71198789 minus 28911987121296
119862 = 051198621199090(1 + 120590) (
119877119899
119877
)
2
(6)
Shock and Vibration 5
1 2 3 4 5minus010
minus005
0
005
010
t (s)
z(m
)
(a) Vertical position
0 1 2 3 4 5minus15
minus10
minus05
0
05
10
15
t (s)
w(m
middotsminus1)
(b) Transverse speed
0 1 2 3 4 5minus0010
minus0005
0
0005
0010
t (s)
120579(r
ad)
(c) Pitch angle
0 1 2 3 4 5minus4
minus2
0
2
4
6
8
t (s)
q(r
ads
)
(d) Pitch rate
1 2 3 4 5minus1
0
1
2
3
t (s)
ℎp
(m)
(e) Immersion depth
0 1 2 3 4 5minus100
0
100
200
300
t (s)
Fp
(N)
(f) Planing force
Figure 8 The motion state of the system when 119896 = minus2195
The feedback controller is designed for the supercavi-tating vehicle with control inputs being the deflection 120575
119890of
the fin and the deflection 120575119888of the cavitator 120575
119890= 119896119911 and
120575119888= 15119911 minus 30120579 minus 03119902 [12 14] were adopted in this paper
where 119896 is the feedback gain of the control variable 119911
3 Dynamic Behavior of the UnderwaterSupercavitating Vehicle
According to [15] the system parameters of the supercavi-tation vehicle are as follows 119892 = 981ms2 119898 = 2 119877
the stablemovement of the supercavitating vehicle the effectsof cavitation number 120590 and the fin control law 119896 on thestable movement state of the vehicle were analyzed basedon the four-dimensional dynamical system Here the restof the parameters remain constant and the two-dimensionalbifurcation diagram (120590 119896) is presented in Figure 2 where theparameters are the cavitation number 120590 and the control gain119896 of the fin deflection angle 120575
119890
6 Shock and Vibration
minus06 minus04 minus02 0 02 04
08
10
12
14
16
18
20
q (rads)
w(m
middotsminus1)
(a)
0 5 10 15minus100
minus80
minus60
minus40
minus20
0
20
t (s)
Lyap
unov
expo
nent
spec
trum
(b)
Figure 9 (a) Phase trajectory diagram in 119908 minus 120579 plane when 119896 = 5 and (b) Lyapunov exponent spectrum
In the phase space of (120590 119896) the dynamic behavior of thesystem is presented in Figure 2 The horizontal section isthe bifurcation diagram of the system for different cavitationnumbers 120590 and the vertical section is the bifurcation diagramof the system when the control gain 119896 varies The parameterranges for different system states can be determined bythe two-dimensional bifurcation diagram The region in redrepresents the stable movement state of the vehicle whichmeans the vehicle will navigate steadily when 120590 and 119896 areequal to the values corresponding to any point (120590 119896) withinthis region The green area shows the periodic oscillatorynature of the vehiclemovement whichmeans that the vehiclewill oscillate periodically and hence will become unstableMoreover the vehicle navigating with the states of the yellowarea will suffer from vibration and impact and then collapseWhen the vehicle alters from the steady state to the periodicstate the Hopf bifurcation occurs The boundary betweenthe red and green regions that is the critical switching lineof the stable state and the periodic state is also called theHopf bifurcation line Similarly the boundary between thegreen and yellow areas indicates the switch between theperiodic and chaotic states where the physical phenomenasuch as tangent bifurcation and period doubling bifurcationcan occur
It can be observed from (1) that in the four-dimensionaldynamical system of the underwater vehicle only the planingforce 119865planing is the nonlinear force associated with thesystem state variable and the vertical velocity 119908 This isprimarily attributed to the fact that the complicated nonlinearforce acts on the fin of the vehicle that the vehicle suffersfrom vibration impact and even collapse due to unstablemovement Therefore the nonlinear dynamic characteristicscan be further understood by analyzing the system from thepoint of view of nonlinearity thus preparing for the stablecontrol of the supercavitating vehicle
31 Nonlinear Dynamical Characteristic of the Vehicle underDifferent Cavitation Values According to the dynamicalbehavior distribution diagram presented in Figure 2 thebifurcation diagram between the system state variable 119908 and
the cavitation number 120590 is provided in Figure 3 (119896 = 1 ie120575119890= 119911 and 120575
119888= 15119911 minus 30120579 minus 03119902) Some simple explanations
are given as followsWhen the cavitation number 120590 of the system falls in
the range of [00198 002687] the trajectory of the vehicleconverges to a stable equilibrium point
When 120590 is equal to 002687 the Hopf bifurcation occursas a result of which the stable equilibrium point becomes thestable periodic trajectory and the vehicle oscillates periodi-cally
After a series of period doubling bifurcation the systemfalls into a chaotic state and the vehicle suffers from hugeimpact When the cavitation number 120590 is approximately003083 the system has three stable periodic trajectories
The bifurcation diagram shown in Figure 3 when 120590 isin
[00315 00325] is magnified in Figure 4 which depicts thediversified bifurcation behaviors of the system
After a series of period doubling bifurcations the systemshifts from three periodic trajectories into three huge chaoticattractors respectively when 120590 is approximately 003197 thisphenomenon is referred to as chaos crisis [12]
When 120590 is equal to 0032 the chaotic attractors suddenlychange into periodic trajectories and form one period-2window and two period-3 windows This phenomenon isreferred to as tangent bifurcation The tangent bifurcationwill cause intermittent chaos and the periodic trajectoriessuddenly develop chaotic bands in the periodic window afterexperiencing a period doubling bifurcation
When 120590 is equal to 003204 the secondary chaotic bandcoincides with the unstable periodic trajectories which thencauses the chaotic crisis The secondary narrow chaotic bandwill then transform into a broad chaotic band
With the increase of 120590 the obvious period-2 windowoccurs for 120590 isin [003207 003225] and when 120590 is approxi-mately 0032228 the broad chaotic band suddenly changesinto two periodic trajectories
32 Nonlinear Dynamic Characteristic of the Vehicle underDifferent Fin Deflection Angles When the cavitation number
Shock and Vibration 7
10 15 200044
0045
0046
0047
0048
z(m
)
t (s)
(a) Vertical position
10 15 2010
15
20
t (s)
w(m
middotsminus1)
(b) Transverse speed
10 15 200020
0021
0022
t (s)
120579(r
ad)
(c) Pitch angle
10 15 20minus03
minus02
minus01
0
01
02
03
q(r
ads
)
t (s)
(d) Pitch rate
minus002
0
002
004
006
008
15 2010t (s)
hp
(m)
(e) Immersion depth
10 15 20minus20
0
20
40
60
80
t (s)
Fp
(N)
(f) Planing force
Figure 10 The motion state of the system when 119896 = 5
is set as 120590 = 00315 the cavitator deflection angle is 120575119888=
15119911 minus 30120579 minus 03119902 and the fin deflection angle is 120575119890= 119896119911 The
bifurcation diagram of the supercavitating vehicle betweenthe system state variable 119908 and the control gain 119896 of the findeflection angle is presented in Figure 5 According to thedynamical behavior distribution presented in Figure 2 the
effective range of the control gain 119896 for the supercavitatingvehicle is presented in Figure 5 when 120590 = 00315
Figure 5 shows that when 119896 falls within the wider range[minus7614 minus5238] the system is in a chaotic state and finallythe period-2 trajectory occurs through period doublingbifurcation
8 Shock and Vibration
minus04 minus02 0 02 04 06minus20
minus10
0
10
20
q (rads)
w(m
middotsminus1)
(a)
0 5 10 15
minus100
minus80
minus60
minus40
minus20
0
20
Lyap
unov
expo
nent
spec
trum
t (s)
(b)
Figure 11 (a) Phase trajectory diagram in 119908 minus 120579 plane when 119896 = minus55 and (b) Lyapunov exponent spectrum
When 119896 is approximately 856 the periodic state endsand the system is in the divergent state The correspondingmagnified part in Figure 5 for minus80 lt 119896 lt minus50 is givenin Figure 6 which indicates that the system changes from aperiodic state into a chaotic state when 119896 = minus779
When 119896 is approximately minus7754 the tangent bifurcationoccurs leading to an intermittent chaos and forming theperiod-3 windows and then three stable periodic trajectories
The period doubling bifurcations occur for the threetrajectories when 119896 is approximately minus77 minus7616 or minus755When 119896 is approximately minus746 the secondary chaotic bandand the unstable periodic trajectories converge into chaos
When 119896 is approximately minus7358 the tangent bifurcationoccurs The system suddenly switches from chaotic state toperiodic state and the period-three windows forms With theoccurrence of a series of period doubling bifurcation thesystem enters into the chaotic state again when 119896 is betweenminus7118 andminus5264 uponwhich the system switches back fromthe chaotic state to the periodic state
4 Movement Characteristic Analysis of theUnderwater Supercavitating Vehicle
When the system of the supercavitating vehicle is not con-trolled the movement state of the system is unstable [12 14]To investigate its movement characteristics alone accordingto the two-dimensional bifurcation diagram the rest of theparameters of the system should be kept constant Assumingthat 120590 is equal to 00315 the feedback control laws 120575
119888=
15119911 minus 30120579 minus 03119902 and 120575119890= minus2195119911 which corresponds to
the point (00315 minus2195) in the red stable movement area inFigure 2Thephase trajectory diagram is shown in Figure 7(a)when the control gain of the fin deflection angle 119896 is equal tominus2195
It can be observed that when 119896 = minus2195 the phasetrajectory of the system gradually stabilizes at an equilibriumpointThe Lyapunov exponent spectrum as a function of time
is presented in Figure 7(b) in which the values of the largestLyapunov exponent curve are negative within a finite time
The motion state of the supercavitating vehicle is pre-sented in Figure 8 inwhich the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate 119902 are attracted to the equilibrium point(00047 00866 00012 0) with less settling time under thecontrol of the law of stable movement
Figures 8(e) and 8(f) demonstrate that the immersiondepth ℎ
1015840 of the fin and the corresponding planing force 119865119901
are both 0 This indicates that the fin is inside the cavity anddoes not have any contact with the cavity the vehicle is in astable navigation state
When 120590 is equal to 00315 the feedback control laws are120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= 3119911 which corresponds to the
point (00315 3) in the green periodic oscillation region inFigure 2
Figure 9(a) tells us that the phase trajectory is a limitcycle with period 2 when the control gain of the fin deflectionangle 119896 is equal to 3 The Lyapunov exponent spectrumcorresponding to 119896 = 3 is shown in Figure 9(b) It is notdifficult to find that the system approximately has a zeroLyapunov exponent and three negative Lyapunov exponents
The motion state of the supercavitating vehicle is shownin Figure 10 in which the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate velocity 119902 oscillate periodically at theequilibrium point (00416 13937 00190 0)
The immersion depth ℎ1015840 of the fin oscillates periodically
in the range of [0 006] (in m) which indicates that thevehicle continuously collides with the cavity wall
The fin is inside the cavity at times and does not come incontact with the cavity which results in zero planing force119865119901 The fin penetrates the cavity into the water at times
and produces the planing force oscillating periodically in therange of [0 48] (in N) The above actions repeat again andagain and such phenomenon is referred to as ldquofin attackphenomenonrdquo It also indicates that the vehicle is in anunstable periodic oscillating state
Shock and Vibration 9
0 05 10minus006
minus004
minus002
0
002
004
006
008
010
z(m
)
t (s)
(a) Vertical position
0 05 10t (s)
minus3
minus2
minus1
0
1
2
3
w(m
middotsminus1)
(b) Transverse speed
0 05 10minus0004
minus0002
0
0002
0004
t (s)
120579(r
ad)
(c) Pitch angle
0 05 10minus10
minus05
0
05
10
t (s)
q(r
ads
)
(d) Pitch rate
0 05 10minus002
0
002
004
006
008
t (s)
hp
(m)
(e) Immersion depth
0 05 10minus60
minus40
minus20
0
20
40
60
80
t (s)
Fp
(N)
(f) Planing force
Figure 12 The motion state of the system when 119896 = 55
When 120590 is equal to 00315 the feedback control laws of120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= minus55119911 are selected which
corresponds to the point (00315 minus55) in the yellow chaoticarea in Figure 2
Figure 11(a) demonstrates that the phase trajectory is achaotic attractor when the control gain of the fin deflectionangle 119896 is equal to minus55 which indicates that the chaoshas occurred and the movement of the vehicle has the
characteristics of a nonlinear dynamic behavior The Lya-punov exponent spectrum corresponding to 119896 = minus55 isgiven in Figure 11(b) It is relatively easy to find that thesystem has a positive Lyapunov exponent and three negativeLyapunov exponents suggesting that the system is in a four-dimensional chaotic state
The motion state of the system is presented in Figure 12After the launch of the supercavitating vehicle the four
10 Shock and Vibration
system state variables namely 119911119908 120579 and 119902 are in the intensenonperiodic oscillating stateThus the vehicle in motion willexperience instability
It can be observed fromFigures 12(e) and 12(f) that withinthe time range of [1 2] (values in 119904) the fin of the vehicleis continuously into contact with the cavity wall under theaction of gravity and produces the planing forceThe planingforce increases gradually with the increase of immersiondepth and the vehicle will rebound inside the cavity whichresults in the loss of the planing force The above actionsrepeat subsequently and the planing force oscillates Theexistence of the planing force will cause vibration and impactto the vehicle resulting in the loss of stability of the vehicleTherefore precise control must be exerted on the vehicle toavoid the above situations [16 17]
5 Conclusions
Thenonlinear dynamic characteristicmovement states underdifferent control parameters of the supercavitating vehiclewere analyzed based on a four-dimensional dynamical modelof the vehicle The following conclusions have been mainlyderived
(1) The movement trajectories of the supercavitatingvehicle have complicated dynamical behavior thesystem will experience Hopf bifurcation periodicalwindows chaos and other nonlinear phenomenawhen the control parameters vary
(2) The movement state of the vehicle under differentcontrol parameters was numerically and preciselyanalyzed according to the phase trajectory diagramthe bifurcation diagram and the Lyapunov exponen-tial spectrum
(3) Most importantly the authors were the first tofind that the range of parameters of the vehicle inany movement state can be determined by a two-dimensional bifurcation diagram The importance ofselecting appropriate control parameters to realize thestable navigation of the supercavitating vehicle wasdemonstrated
It is believed that the work presented in this paper is ofgreat importance for further studies on the stable controlof the underwater supercavitating vehicles especially forengineering practice
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 11472163 11402116 and 61473340)
References
[1] G Wang and M Ostoja-Starzewski ldquoLarge eddy simulationof a sheetcloud cavitation on a NACA0015 hydrofoilrdquo AppliedMathematical Modelling vol 31 no 3 pp 417ndash447 2007
[2] Q T Li Y S He and L P Xue ldquoA numerical simulationof pitching motion of the ventilated supercaviting vehicleapproximately its frontrdquo Chinese Journal of Hydrodynamics vol26 no 6 pp 589ndash685 2011
[3] A K Singhal M M Athavale H Li and Y Jiang ldquoMathemati-cal basis and validation of the full cavitation modelrdquo Journal ofFluids Engineering vol 124 no 3 pp 617ndash624 2002
[4] Y N Savchenko ldquoSupercavitation problems and perspectivesrdquoin Proceedings of the 4th International Symposium onCavitationPasadena Calif USA April 2001
[5] M A Hassouneh and E H Abed ldquoLyapunov and LMI analysisand feedback control of border collision bifurcationsrdquo Nonlin-ear Dynamics vol 50 no 3 pp 373ndash386 2007
[6] Y J Wei J H Wang J Z Zhang W Cao and W H HuangldquoNonlinear dynamics and control of underwater supercavitat-ing vehiclerdquo Journal of Vibration And Shock vol 28 no 6 pp179ndash204 2009
[7] S S Kulkarni and R Pratap ldquoStudies on the dynamics ofa supercavitating projectilerdquo Applied Mathematical Modellingvol 24 no 2 pp 113ndash129 2000
[8] J-Y Choi M Ruzzene and O A Bauchau ldquoDynamic anal-ysis of flexible supercavitating vehicles using modal-basedelementsrdquo Simulation vol 80 no 11 pp 619ndash633 2004
[9] B Feeny ldquoA nonsmooth Coulomb friction oscillatorrdquo Physica DNonlinear Phenomena vol 59 no 1ndash3 pp 25ndash38 1992
[10] A D Vasin and E V Paryshev ldquoImmersion of cylinder in a fluidthrough a cylindrical free surfacerdquo Fluid Dynamics vol 36 no2 pp 169ndash177 2001
[11] G Lin B Balachandran and E Abed ldquoSupercavitating bodydynamics bifurcations and controlrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 691ndash696 PortlandOre USA June 2005
[12] J Dzielski and A Kurdila ldquoA benchmark control problemfor supercavitating vehicles and an initial investigation ofsolutionsrdquo Journal of Vibration amp Control vol 9 no 7 pp 791ndash804 2003
[13] G J Lin B Balakumar and H A Eysd ldquoBifurcation behaviorof a supercavitating vehiclerdquo in Proceedings of the ASMEInternational Mechanical Engineering Congress and Expositionpp 293ndash300 Chicago Ill USA November 2006
[14] G J Lin B Balachandran and E H Abed ldquoDynamics andcontrol of supercavitating vehiclesrdquo Journal of Dynamic SystemsMeasurement and Control vol 130 no 2 Article ID 021003 pp281ndash287 2008
[15] M A Hassouneh V Nguyen B Balachandran and E HAbed ldquoStability analysis and control of supercavitating vehicleswith advection delayrdquo Journal of Computational and NonlinearDynamics vol 8 no 2 Article ID 021003 2012
[16] G Lin B Balachandran and E H Abed ldquoNonlinear dynamicsand bifurcations of a supercavitating vehiclerdquo IEEE Journal ofOceanic Engineering vol 32 no 4 pp 753ndash761 2007
[17] J H Wang Y J Wei and K P Yu ldquoModeling and control ofunderwater supercavitating vehicle based on memory effect ofcavityrdquo Journal of Vibration And Shock vol 29 no 8 pp 160ndash163 2010
contact between the fins and the cavity walls The mostimportant finding of this study was that the region andparameter range of the vehicle in a stable movement stateare determined by the two-dimensional bifurcation diagramThe movement characteristics of the supercavitation vehicleunder different control parameters were also discussed indetail
To the best of our knowledge it is very difficult to find anyrelated work in this paper up till now
2 Dynamic Modeling ofthe Supercavitating Vehicle
21 Force Analysis When the underwater vehicle is navi-gating at high speed in the supercavitating state most ofthe vehicle will be enveloped by the cavity and only asmall part of the surface will have a contact with the waterWhile the cavitator at the front has a direct contact withwater the cavitator can rotate by a certain angle Differenthydrodynamic power can be provided for the vehicle withthe change of the angle and a planing force can be producedwhen the fin contacts with the cavity wall Four fins weresymmetrically arranged in the rear part of the vehicle Apart of the fin penetrates the cavity wall to have a directcontact with the water thus providing the required forceand momentum to stabilize and control the vehicle with thecavitator In this case the control surface is composed of thecavitator and the four fins [5] The deflection angles of thecavitator and the fins are usually selected as the feedbackcontrol inputs to ensure the stable underwater movementof the vehicle The shape and the force diagram of thesupercavitating vehicle are presented in Figure 1
The forces acting on the vehicle in its own coordinatesystem are indicated in Figure 1 The main forces include thelift force on the cavitator 119865cavitator the lift force on the fin119865fins the gravity at the centroid of the vehicle 119865gravity andthe planing force generated by the interaction between the finand the cavity wall 119865planing the last of which is a complicatednonlinear planing force that consequently causes vibrationand impact to the vehicleThe expression of the planing forceis as follows [11]
119865planing = minus1198812
[1 minus (
1198771015840
ℎ1015840+ 1198771015840)
2
](
1 + ℎ1015840
1 + 2ℎ1015840)120572 (1)
where 119881 is the velocity of the vehicle and 1198771015840
= (119877119888minus
119877)119877 where 119877119888and 119877 are the radius of the cavity and the
vehicle respectively The immersion depth of the aft of thesupercavitating vehicle ℎ1015840 is given as follows [11]
+ (119908 minus 1199081199050) tanh [119896 (119908 minus 119908
1199050)]
(3)
Fin w
q
V
Ffin
FplaningFgravity Cavitator
FCavitator
Figure 1 Shape and force diagram of the supercavitating vehicle
0020 0024 0028 0032 0036minus60
0
20
40
minus20
minus40
120590
k
Figure 2 Dynamic behavior distribution diagram of the system
where the positive value of 119908 at the transition point is1199081199050
= (119877119888minus 119877)119881119871 and 119896 is a constant used to control the
approximated error which is generally set to 300The geometrical angle between the vehicle centerline
and the cavity centerline is the immersion angle 120572 of thesupercavitating vehicle expressed as [11]
120572 =
119908
119881
minus tanh (119896119908)119888
119881
(4)
where 119877119888is the cavity radius and
119888is the shrinkage ratio at a
distance 119871 from the cavitator
22 Dynamic Modeling Through the interactive relationshipbetween the vehicle and the cavity obtained in the previoussection the model can be established based on the forceequations [12] According to the coordinate system presentedin [13] the origin is the center of the disk cavitator at thefront of the supercavitating vehicle 119883-axis aligns with thesymmetry axis of the vehicle and points forward 119885-axis isperpendicular to119883-axis facing vertically downward and119908 isthe velocity in119885-axis direction119881 represents the longitudinalvelocity of the cavitator at the vehicle front and 120579 119902 and 119911
are the pitching angle pitching rate and depth of the vehiclerespectively
Shock and Vibration 3
120590
w
40
30
20
10
000360032002800240020
Figure 3 Bifurcation diagram varying with 120590 when 119896 = 1
13
12
11
10
09
0032500323003210031900317
120590
w
Figure 4 Bifurcation diagram for 00315 lt 120590 lt 00325
According to the theory of rigid body dynamics thefollowing relationship can be derived relating the abovevariables [13]
(
120579
) = (
0 1 minus119881 0
0 11988622
0 11988624
0 0 0 1
0 11988642
0 11988644
)(
119911
119908
120579
119902
)
+(
0 0
11988721
11988722
0 0
11988741
11988742
)(
120575119890
120575119888
) +(
0
1198882
0
0
)
+(
0
1198892
0
1198894
)119865planing
(5)
where
11988622=
119862119881119879
119898
(
minus1 minus 119899
119871
) 119878 +
17
36
119899119871
11988624= 119881119879119878 (119862
minus119899
119898
+
7
9
) minus 119881119879(119862
minus119899
119898
+
17
36
)
17
36
1198712
11988642=
119862119881119879
119898
(
17
36
minus
11119899
36
)
11988644=
minus11119862119881119879119899119871
36119898
11988721=
1198621198812
119879119899
119898
(
minus119878
119871
+
17119871
36
)
11988722=
minus1198621198812
119879119878
119898119871
11988741=
minus111198621198812
119879119899
36119898
11988742=
171198621198812
119879
36119898
1198882= 119892
4 Shock and Vibration
30
20
0
minus10
minus20200minus20minus40minus60minus80
k
w
10
Figure 5 Bifurcation diagram varying with 119896 when 120590 = 00315
15
10
05
0
minus05
k
w
minus50minus55minus60minus65minus70minus75minus80
Figure 6 Bifurcation diagram for minus80 lt 119896 lt minus50
minus0001 0 0001006
007
008
009
010
Equilibrium point
q (rads )
w(m
middotsminus1)
(a)
0 5 10 15
minus100
minus80
minus60
minus40
minus20
0
20
Lyap
unov
expo
nent
spec
trum
t (s)
(b)
Figure 7 (a) Phase trajectory diagram in 119908 minus 120579 plane and (b) Lyapunov exponent spectrum
1198892=
119879
119898
(
minus17119871
36
+
119878
119871
)
1198894=
11119879
36119898
119878 =
11
60
1198772
+
1331198712
405
119879 =
1
71198789 minus 28911987121296
119862 = 051198621199090(1 + 120590) (
119877119899
119877
)
2
(6)
Shock and Vibration 5
1 2 3 4 5minus010
minus005
0
005
010
t (s)
z(m
)
(a) Vertical position
0 1 2 3 4 5minus15
minus10
minus05
0
05
10
15
t (s)
w(m
middotsminus1)
(b) Transverse speed
0 1 2 3 4 5minus0010
minus0005
0
0005
0010
t (s)
120579(r
ad)
(c) Pitch angle
0 1 2 3 4 5minus4
minus2
0
2
4
6
8
t (s)
q(r
ads
)
(d) Pitch rate
1 2 3 4 5minus1
0
1
2
3
t (s)
ℎp
(m)
(e) Immersion depth
0 1 2 3 4 5minus100
0
100
200
300
t (s)
Fp
(N)
(f) Planing force
Figure 8 The motion state of the system when 119896 = minus2195
The feedback controller is designed for the supercavi-tating vehicle with control inputs being the deflection 120575
119890of
the fin and the deflection 120575119888of the cavitator 120575
119890= 119896119911 and
120575119888= 15119911 minus 30120579 minus 03119902 [12 14] were adopted in this paper
where 119896 is the feedback gain of the control variable 119911
3 Dynamic Behavior of the UnderwaterSupercavitating Vehicle
According to [15] the system parameters of the supercavi-tation vehicle are as follows 119892 = 981ms2 119898 = 2 119877
the stablemovement of the supercavitating vehicle the effectsof cavitation number 120590 and the fin control law 119896 on thestable movement state of the vehicle were analyzed basedon the four-dimensional dynamical system Here the restof the parameters remain constant and the two-dimensionalbifurcation diagram (120590 119896) is presented in Figure 2 where theparameters are the cavitation number 120590 and the control gain119896 of the fin deflection angle 120575
119890
6 Shock and Vibration
minus06 minus04 minus02 0 02 04
08
10
12
14
16
18
20
q (rads)
w(m
middotsminus1)
(a)
0 5 10 15minus100
minus80
minus60
minus40
minus20
0
20
t (s)
Lyap
unov
expo
nent
spec
trum
(b)
Figure 9 (a) Phase trajectory diagram in 119908 minus 120579 plane when 119896 = 5 and (b) Lyapunov exponent spectrum
In the phase space of (120590 119896) the dynamic behavior of thesystem is presented in Figure 2 The horizontal section isthe bifurcation diagram of the system for different cavitationnumbers 120590 and the vertical section is the bifurcation diagramof the system when the control gain 119896 varies The parameterranges for different system states can be determined bythe two-dimensional bifurcation diagram The region in redrepresents the stable movement state of the vehicle whichmeans the vehicle will navigate steadily when 120590 and 119896 areequal to the values corresponding to any point (120590 119896) withinthis region The green area shows the periodic oscillatorynature of the vehiclemovement whichmeans that the vehiclewill oscillate periodically and hence will become unstableMoreover the vehicle navigating with the states of the yellowarea will suffer from vibration and impact and then collapseWhen the vehicle alters from the steady state to the periodicstate the Hopf bifurcation occurs The boundary betweenthe red and green regions that is the critical switching lineof the stable state and the periodic state is also called theHopf bifurcation line Similarly the boundary between thegreen and yellow areas indicates the switch between theperiodic and chaotic states where the physical phenomenasuch as tangent bifurcation and period doubling bifurcationcan occur
It can be observed from (1) that in the four-dimensionaldynamical system of the underwater vehicle only the planingforce 119865planing is the nonlinear force associated with thesystem state variable and the vertical velocity 119908 This isprimarily attributed to the fact that the complicated nonlinearforce acts on the fin of the vehicle that the vehicle suffersfrom vibration impact and even collapse due to unstablemovement Therefore the nonlinear dynamic characteristicscan be further understood by analyzing the system from thepoint of view of nonlinearity thus preparing for the stablecontrol of the supercavitating vehicle
31 Nonlinear Dynamical Characteristic of the Vehicle underDifferent Cavitation Values According to the dynamicalbehavior distribution diagram presented in Figure 2 thebifurcation diagram between the system state variable 119908 and
the cavitation number 120590 is provided in Figure 3 (119896 = 1 ie120575119890= 119911 and 120575
119888= 15119911 minus 30120579 minus 03119902) Some simple explanations
are given as followsWhen the cavitation number 120590 of the system falls in
the range of [00198 002687] the trajectory of the vehicleconverges to a stable equilibrium point
When 120590 is equal to 002687 the Hopf bifurcation occursas a result of which the stable equilibrium point becomes thestable periodic trajectory and the vehicle oscillates periodi-cally
After a series of period doubling bifurcation the systemfalls into a chaotic state and the vehicle suffers from hugeimpact When the cavitation number 120590 is approximately003083 the system has three stable periodic trajectories
The bifurcation diagram shown in Figure 3 when 120590 isin
[00315 00325] is magnified in Figure 4 which depicts thediversified bifurcation behaviors of the system
After a series of period doubling bifurcations the systemshifts from three periodic trajectories into three huge chaoticattractors respectively when 120590 is approximately 003197 thisphenomenon is referred to as chaos crisis [12]
When 120590 is equal to 0032 the chaotic attractors suddenlychange into periodic trajectories and form one period-2window and two period-3 windows This phenomenon isreferred to as tangent bifurcation The tangent bifurcationwill cause intermittent chaos and the periodic trajectoriessuddenly develop chaotic bands in the periodic window afterexperiencing a period doubling bifurcation
When 120590 is equal to 003204 the secondary chaotic bandcoincides with the unstable periodic trajectories which thencauses the chaotic crisis The secondary narrow chaotic bandwill then transform into a broad chaotic band
With the increase of 120590 the obvious period-2 windowoccurs for 120590 isin [003207 003225] and when 120590 is approxi-mately 0032228 the broad chaotic band suddenly changesinto two periodic trajectories
32 Nonlinear Dynamic Characteristic of the Vehicle underDifferent Fin Deflection Angles When the cavitation number
Shock and Vibration 7
10 15 200044
0045
0046
0047
0048
z(m
)
t (s)
(a) Vertical position
10 15 2010
15
20
t (s)
w(m
middotsminus1)
(b) Transverse speed
10 15 200020
0021
0022
t (s)
120579(r
ad)
(c) Pitch angle
10 15 20minus03
minus02
minus01
0
01
02
03
q(r
ads
)
t (s)
(d) Pitch rate
minus002
0
002
004
006
008
15 2010t (s)
hp
(m)
(e) Immersion depth
10 15 20minus20
0
20
40
60
80
t (s)
Fp
(N)
(f) Planing force
Figure 10 The motion state of the system when 119896 = 5
is set as 120590 = 00315 the cavitator deflection angle is 120575119888=
15119911 minus 30120579 minus 03119902 and the fin deflection angle is 120575119890= 119896119911 The
bifurcation diagram of the supercavitating vehicle betweenthe system state variable 119908 and the control gain 119896 of the findeflection angle is presented in Figure 5 According to thedynamical behavior distribution presented in Figure 2 the
effective range of the control gain 119896 for the supercavitatingvehicle is presented in Figure 5 when 120590 = 00315
Figure 5 shows that when 119896 falls within the wider range[minus7614 minus5238] the system is in a chaotic state and finallythe period-2 trajectory occurs through period doublingbifurcation
8 Shock and Vibration
minus04 minus02 0 02 04 06minus20
minus10
0
10
20
q (rads)
w(m
middotsminus1)
(a)
0 5 10 15
minus100
minus80
minus60
minus40
minus20
0
20
Lyap
unov
expo
nent
spec
trum
t (s)
(b)
Figure 11 (a) Phase trajectory diagram in 119908 minus 120579 plane when 119896 = minus55 and (b) Lyapunov exponent spectrum
When 119896 is approximately 856 the periodic state endsand the system is in the divergent state The correspondingmagnified part in Figure 5 for minus80 lt 119896 lt minus50 is givenin Figure 6 which indicates that the system changes from aperiodic state into a chaotic state when 119896 = minus779
When 119896 is approximately minus7754 the tangent bifurcationoccurs leading to an intermittent chaos and forming theperiod-3 windows and then three stable periodic trajectories
The period doubling bifurcations occur for the threetrajectories when 119896 is approximately minus77 minus7616 or minus755When 119896 is approximately minus746 the secondary chaotic bandand the unstable periodic trajectories converge into chaos
When 119896 is approximately minus7358 the tangent bifurcationoccurs The system suddenly switches from chaotic state toperiodic state and the period-three windows forms With theoccurrence of a series of period doubling bifurcation thesystem enters into the chaotic state again when 119896 is betweenminus7118 andminus5264 uponwhich the system switches back fromthe chaotic state to the periodic state
4 Movement Characteristic Analysis of theUnderwater Supercavitating Vehicle
When the system of the supercavitating vehicle is not con-trolled the movement state of the system is unstable [12 14]To investigate its movement characteristics alone accordingto the two-dimensional bifurcation diagram the rest of theparameters of the system should be kept constant Assumingthat 120590 is equal to 00315 the feedback control laws 120575
119888=
15119911 minus 30120579 minus 03119902 and 120575119890= minus2195119911 which corresponds to
the point (00315 minus2195) in the red stable movement area inFigure 2Thephase trajectory diagram is shown in Figure 7(a)when the control gain of the fin deflection angle 119896 is equal tominus2195
It can be observed that when 119896 = minus2195 the phasetrajectory of the system gradually stabilizes at an equilibriumpointThe Lyapunov exponent spectrum as a function of time
is presented in Figure 7(b) in which the values of the largestLyapunov exponent curve are negative within a finite time
The motion state of the supercavitating vehicle is pre-sented in Figure 8 inwhich the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate 119902 are attracted to the equilibrium point(00047 00866 00012 0) with less settling time under thecontrol of the law of stable movement
Figures 8(e) and 8(f) demonstrate that the immersiondepth ℎ
1015840 of the fin and the corresponding planing force 119865119901
are both 0 This indicates that the fin is inside the cavity anddoes not have any contact with the cavity the vehicle is in astable navigation state
When 120590 is equal to 00315 the feedback control laws are120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= 3119911 which corresponds to the
point (00315 3) in the green periodic oscillation region inFigure 2
Figure 9(a) tells us that the phase trajectory is a limitcycle with period 2 when the control gain of the fin deflectionangle 119896 is equal to 3 The Lyapunov exponent spectrumcorresponding to 119896 = 3 is shown in Figure 9(b) It is notdifficult to find that the system approximately has a zeroLyapunov exponent and three negative Lyapunov exponents
The motion state of the supercavitating vehicle is shownin Figure 10 in which the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate velocity 119902 oscillate periodically at theequilibrium point (00416 13937 00190 0)
The immersion depth ℎ1015840 of the fin oscillates periodically
in the range of [0 006] (in m) which indicates that thevehicle continuously collides with the cavity wall
The fin is inside the cavity at times and does not come incontact with the cavity which results in zero planing force119865119901 The fin penetrates the cavity into the water at times
and produces the planing force oscillating periodically in therange of [0 48] (in N) The above actions repeat again andagain and such phenomenon is referred to as ldquofin attackphenomenonrdquo It also indicates that the vehicle is in anunstable periodic oscillating state
Shock and Vibration 9
0 05 10minus006
minus004
minus002
0
002
004
006
008
010
z(m
)
t (s)
(a) Vertical position
0 05 10t (s)
minus3
minus2
minus1
0
1
2
3
w(m
middotsminus1)
(b) Transverse speed
0 05 10minus0004
minus0002
0
0002
0004
t (s)
120579(r
ad)
(c) Pitch angle
0 05 10minus10
minus05
0
05
10
t (s)
q(r
ads
)
(d) Pitch rate
0 05 10minus002
0
002
004
006
008
t (s)
hp
(m)
(e) Immersion depth
0 05 10minus60
minus40
minus20
0
20
40
60
80
t (s)
Fp
(N)
(f) Planing force
Figure 12 The motion state of the system when 119896 = 55
When 120590 is equal to 00315 the feedback control laws of120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= minus55119911 are selected which
corresponds to the point (00315 minus55) in the yellow chaoticarea in Figure 2
Figure 11(a) demonstrates that the phase trajectory is achaotic attractor when the control gain of the fin deflectionangle 119896 is equal to minus55 which indicates that the chaoshas occurred and the movement of the vehicle has the
characteristics of a nonlinear dynamic behavior The Lya-punov exponent spectrum corresponding to 119896 = minus55 isgiven in Figure 11(b) It is relatively easy to find that thesystem has a positive Lyapunov exponent and three negativeLyapunov exponents suggesting that the system is in a four-dimensional chaotic state
The motion state of the system is presented in Figure 12After the launch of the supercavitating vehicle the four
10 Shock and Vibration
system state variables namely 119911119908 120579 and 119902 are in the intensenonperiodic oscillating stateThus the vehicle in motion willexperience instability
It can be observed fromFigures 12(e) and 12(f) that withinthe time range of [1 2] (values in 119904) the fin of the vehicleis continuously into contact with the cavity wall under theaction of gravity and produces the planing forceThe planingforce increases gradually with the increase of immersiondepth and the vehicle will rebound inside the cavity whichresults in the loss of the planing force The above actionsrepeat subsequently and the planing force oscillates Theexistence of the planing force will cause vibration and impactto the vehicle resulting in the loss of stability of the vehicleTherefore precise control must be exerted on the vehicle toavoid the above situations [16 17]
5 Conclusions
Thenonlinear dynamic characteristicmovement states underdifferent control parameters of the supercavitating vehiclewere analyzed based on a four-dimensional dynamical modelof the vehicle The following conclusions have been mainlyderived
(1) The movement trajectories of the supercavitatingvehicle have complicated dynamical behavior thesystem will experience Hopf bifurcation periodicalwindows chaos and other nonlinear phenomenawhen the control parameters vary
(2) The movement state of the vehicle under differentcontrol parameters was numerically and preciselyanalyzed according to the phase trajectory diagramthe bifurcation diagram and the Lyapunov exponen-tial spectrum
(3) Most importantly the authors were the first tofind that the range of parameters of the vehicle inany movement state can be determined by a two-dimensional bifurcation diagram The importance ofselecting appropriate control parameters to realize thestable navigation of the supercavitating vehicle wasdemonstrated
It is believed that the work presented in this paper is ofgreat importance for further studies on the stable controlof the underwater supercavitating vehicles especially forengineering practice
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 11472163 11402116 and 61473340)
References
[1] G Wang and M Ostoja-Starzewski ldquoLarge eddy simulationof a sheetcloud cavitation on a NACA0015 hydrofoilrdquo AppliedMathematical Modelling vol 31 no 3 pp 417ndash447 2007
[2] Q T Li Y S He and L P Xue ldquoA numerical simulationof pitching motion of the ventilated supercaviting vehicleapproximately its frontrdquo Chinese Journal of Hydrodynamics vol26 no 6 pp 589ndash685 2011
[3] A K Singhal M M Athavale H Li and Y Jiang ldquoMathemati-cal basis and validation of the full cavitation modelrdquo Journal ofFluids Engineering vol 124 no 3 pp 617ndash624 2002
[4] Y N Savchenko ldquoSupercavitation problems and perspectivesrdquoin Proceedings of the 4th International Symposium onCavitationPasadena Calif USA April 2001
[5] M A Hassouneh and E H Abed ldquoLyapunov and LMI analysisand feedback control of border collision bifurcationsrdquo Nonlin-ear Dynamics vol 50 no 3 pp 373ndash386 2007
[6] Y J Wei J H Wang J Z Zhang W Cao and W H HuangldquoNonlinear dynamics and control of underwater supercavitat-ing vehiclerdquo Journal of Vibration And Shock vol 28 no 6 pp179ndash204 2009
[7] S S Kulkarni and R Pratap ldquoStudies on the dynamics ofa supercavitating projectilerdquo Applied Mathematical Modellingvol 24 no 2 pp 113ndash129 2000
[8] J-Y Choi M Ruzzene and O A Bauchau ldquoDynamic anal-ysis of flexible supercavitating vehicles using modal-basedelementsrdquo Simulation vol 80 no 11 pp 619ndash633 2004
[9] B Feeny ldquoA nonsmooth Coulomb friction oscillatorrdquo Physica DNonlinear Phenomena vol 59 no 1ndash3 pp 25ndash38 1992
[10] A D Vasin and E V Paryshev ldquoImmersion of cylinder in a fluidthrough a cylindrical free surfacerdquo Fluid Dynamics vol 36 no2 pp 169ndash177 2001
[11] G Lin B Balachandran and E Abed ldquoSupercavitating bodydynamics bifurcations and controlrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 691ndash696 PortlandOre USA June 2005
[12] J Dzielski and A Kurdila ldquoA benchmark control problemfor supercavitating vehicles and an initial investigation ofsolutionsrdquo Journal of Vibration amp Control vol 9 no 7 pp 791ndash804 2003
[13] G J Lin B Balakumar and H A Eysd ldquoBifurcation behaviorof a supercavitating vehiclerdquo in Proceedings of the ASMEInternational Mechanical Engineering Congress and Expositionpp 293ndash300 Chicago Ill USA November 2006
[14] G J Lin B Balachandran and E H Abed ldquoDynamics andcontrol of supercavitating vehiclesrdquo Journal of Dynamic SystemsMeasurement and Control vol 130 no 2 Article ID 021003 pp281ndash287 2008
[15] M A Hassouneh V Nguyen B Balachandran and E HAbed ldquoStability analysis and control of supercavitating vehicleswith advection delayrdquo Journal of Computational and NonlinearDynamics vol 8 no 2 Article ID 021003 2012
[16] G Lin B Balachandran and E H Abed ldquoNonlinear dynamicsand bifurcations of a supercavitating vehiclerdquo IEEE Journal ofOceanic Engineering vol 32 no 4 pp 753ndash761 2007
[17] J H Wang Y J Wei and K P Yu ldquoModeling and control ofunderwater supercavitating vehicle based on memory effect ofcavityrdquo Journal of Vibration And Shock vol 29 no 8 pp 160ndash163 2010
the stablemovement of the supercavitating vehicle the effectsof cavitation number 120590 and the fin control law 119896 on thestable movement state of the vehicle were analyzed basedon the four-dimensional dynamical system Here the restof the parameters remain constant and the two-dimensionalbifurcation diagram (120590 119896) is presented in Figure 2 where theparameters are the cavitation number 120590 and the control gain119896 of the fin deflection angle 120575
119890
6 Shock and Vibration
minus06 minus04 minus02 0 02 04
08
10
12
14
16
18
20
q (rads)
w(m
middotsminus1)
(a)
0 5 10 15minus100
minus80
minus60
minus40
minus20
0
20
t (s)
Lyap
unov
expo
nent
spec
trum
(b)
Figure 9 (a) Phase trajectory diagram in 119908 minus 120579 plane when 119896 = 5 and (b) Lyapunov exponent spectrum
In the phase space of (120590 119896) the dynamic behavior of thesystem is presented in Figure 2 The horizontal section isthe bifurcation diagram of the system for different cavitationnumbers 120590 and the vertical section is the bifurcation diagramof the system when the control gain 119896 varies The parameterranges for different system states can be determined bythe two-dimensional bifurcation diagram The region in redrepresents the stable movement state of the vehicle whichmeans the vehicle will navigate steadily when 120590 and 119896 areequal to the values corresponding to any point (120590 119896) withinthis region The green area shows the periodic oscillatorynature of the vehiclemovement whichmeans that the vehiclewill oscillate periodically and hence will become unstableMoreover the vehicle navigating with the states of the yellowarea will suffer from vibration and impact and then collapseWhen the vehicle alters from the steady state to the periodicstate the Hopf bifurcation occurs The boundary betweenthe red and green regions that is the critical switching lineof the stable state and the periodic state is also called theHopf bifurcation line Similarly the boundary between thegreen and yellow areas indicates the switch between theperiodic and chaotic states where the physical phenomenasuch as tangent bifurcation and period doubling bifurcationcan occur
It can be observed from (1) that in the four-dimensionaldynamical system of the underwater vehicle only the planingforce 119865planing is the nonlinear force associated with thesystem state variable and the vertical velocity 119908 This isprimarily attributed to the fact that the complicated nonlinearforce acts on the fin of the vehicle that the vehicle suffersfrom vibration impact and even collapse due to unstablemovement Therefore the nonlinear dynamic characteristicscan be further understood by analyzing the system from thepoint of view of nonlinearity thus preparing for the stablecontrol of the supercavitating vehicle
31 Nonlinear Dynamical Characteristic of the Vehicle underDifferent Cavitation Values According to the dynamicalbehavior distribution diagram presented in Figure 2 thebifurcation diagram between the system state variable 119908 and
the cavitation number 120590 is provided in Figure 3 (119896 = 1 ie120575119890= 119911 and 120575
119888= 15119911 minus 30120579 minus 03119902) Some simple explanations
are given as followsWhen the cavitation number 120590 of the system falls in
the range of [00198 002687] the trajectory of the vehicleconverges to a stable equilibrium point
When 120590 is equal to 002687 the Hopf bifurcation occursas a result of which the stable equilibrium point becomes thestable periodic trajectory and the vehicle oscillates periodi-cally
After a series of period doubling bifurcation the systemfalls into a chaotic state and the vehicle suffers from hugeimpact When the cavitation number 120590 is approximately003083 the system has three stable periodic trajectories
The bifurcation diagram shown in Figure 3 when 120590 isin
[00315 00325] is magnified in Figure 4 which depicts thediversified bifurcation behaviors of the system
After a series of period doubling bifurcations the systemshifts from three periodic trajectories into three huge chaoticattractors respectively when 120590 is approximately 003197 thisphenomenon is referred to as chaos crisis [12]
When 120590 is equal to 0032 the chaotic attractors suddenlychange into periodic trajectories and form one period-2window and two period-3 windows This phenomenon isreferred to as tangent bifurcation The tangent bifurcationwill cause intermittent chaos and the periodic trajectoriessuddenly develop chaotic bands in the periodic window afterexperiencing a period doubling bifurcation
When 120590 is equal to 003204 the secondary chaotic bandcoincides with the unstable periodic trajectories which thencauses the chaotic crisis The secondary narrow chaotic bandwill then transform into a broad chaotic band
With the increase of 120590 the obvious period-2 windowoccurs for 120590 isin [003207 003225] and when 120590 is approxi-mately 0032228 the broad chaotic band suddenly changesinto two periodic trajectories
32 Nonlinear Dynamic Characteristic of the Vehicle underDifferent Fin Deflection Angles When the cavitation number
Shock and Vibration 7
10 15 200044
0045
0046
0047
0048
z(m
)
t (s)
(a) Vertical position
10 15 2010
15
20
t (s)
w(m
middotsminus1)
(b) Transverse speed
10 15 200020
0021
0022
t (s)
120579(r
ad)
(c) Pitch angle
10 15 20minus03
minus02
minus01
0
01
02
03
q(r
ads
)
t (s)
(d) Pitch rate
minus002
0
002
004
006
008
15 2010t (s)
hp
(m)
(e) Immersion depth
10 15 20minus20
0
20
40
60
80
t (s)
Fp
(N)
(f) Planing force
Figure 10 The motion state of the system when 119896 = 5
is set as 120590 = 00315 the cavitator deflection angle is 120575119888=
15119911 minus 30120579 minus 03119902 and the fin deflection angle is 120575119890= 119896119911 The
bifurcation diagram of the supercavitating vehicle betweenthe system state variable 119908 and the control gain 119896 of the findeflection angle is presented in Figure 5 According to thedynamical behavior distribution presented in Figure 2 the
effective range of the control gain 119896 for the supercavitatingvehicle is presented in Figure 5 when 120590 = 00315
Figure 5 shows that when 119896 falls within the wider range[minus7614 minus5238] the system is in a chaotic state and finallythe period-2 trajectory occurs through period doublingbifurcation
8 Shock and Vibration
minus04 minus02 0 02 04 06minus20
minus10
0
10
20
q (rads)
w(m
middotsminus1)
(a)
0 5 10 15
minus100
minus80
minus60
minus40
minus20
0
20
Lyap
unov
expo
nent
spec
trum
t (s)
(b)
Figure 11 (a) Phase trajectory diagram in 119908 minus 120579 plane when 119896 = minus55 and (b) Lyapunov exponent spectrum
When 119896 is approximately 856 the periodic state endsand the system is in the divergent state The correspondingmagnified part in Figure 5 for minus80 lt 119896 lt minus50 is givenin Figure 6 which indicates that the system changes from aperiodic state into a chaotic state when 119896 = minus779
When 119896 is approximately minus7754 the tangent bifurcationoccurs leading to an intermittent chaos and forming theperiod-3 windows and then three stable periodic trajectories
The period doubling bifurcations occur for the threetrajectories when 119896 is approximately minus77 minus7616 or minus755When 119896 is approximately minus746 the secondary chaotic bandand the unstable periodic trajectories converge into chaos
When 119896 is approximately minus7358 the tangent bifurcationoccurs The system suddenly switches from chaotic state toperiodic state and the period-three windows forms With theoccurrence of a series of period doubling bifurcation thesystem enters into the chaotic state again when 119896 is betweenminus7118 andminus5264 uponwhich the system switches back fromthe chaotic state to the periodic state
4 Movement Characteristic Analysis of theUnderwater Supercavitating Vehicle
When the system of the supercavitating vehicle is not con-trolled the movement state of the system is unstable [12 14]To investigate its movement characteristics alone accordingto the two-dimensional bifurcation diagram the rest of theparameters of the system should be kept constant Assumingthat 120590 is equal to 00315 the feedback control laws 120575
119888=
15119911 minus 30120579 minus 03119902 and 120575119890= minus2195119911 which corresponds to
the point (00315 minus2195) in the red stable movement area inFigure 2Thephase trajectory diagram is shown in Figure 7(a)when the control gain of the fin deflection angle 119896 is equal tominus2195
It can be observed that when 119896 = minus2195 the phasetrajectory of the system gradually stabilizes at an equilibriumpointThe Lyapunov exponent spectrum as a function of time
is presented in Figure 7(b) in which the values of the largestLyapunov exponent curve are negative within a finite time
The motion state of the supercavitating vehicle is pre-sented in Figure 8 inwhich the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate 119902 are attracted to the equilibrium point(00047 00866 00012 0) with less settling time under thecontrol of the law of stable movement
Figures 8(e) and 8(f) demonstrate that the immersiondepth ℎ
1015840 of the fin and the corresponding planing force 119865119901
are both 0 This indicates that the fin is inside the cavity anddoes not have any contact with the cavity the vehicle is in astable navigation state
When 120590 is equal to 00315 the feedback control laws are120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= 3119911 which corresponds to the
point (00315 3) in the green periodic oscillation region inFigure 2
Figure 9(a) tells us that the phase trajectory is a limitcycle with period 2 when the control gain of the fin deflectionangle 119896 is equal to 3 The Lyapunov exponent spectrumcorresponding to 119896 = 3 is shown in Figure 9(b) It is notdifficult to find that the system approximately has a zeroLyapunov exponent and three negative Lyapunov exponents
The motion state of the supercavitating vehicle is shownin Figure 10 in which the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate velocity 119902 oscillate periodically at theequilibrium point (00416 13937 00190 0)
The immersion depth ℎ1015840 of the fin oscillates periodically
in the range of [0 006] (in m) which indicates that thevehicle continuously collides with the cavity wall
The fin is inside the cavity at times and does not come incontact with the cavity which results in zero planing force119865119901 The fin penetrates the cavity into the water at times
and produces the planing force oscillating periodically in therange of [0 48] (in N) The above actions repeat again andagain and such phenomenon is referred to as ldquofin attackphenomenonrdquo It also indicates that the vehicle is in anunstable periodic oscillating state
Shock and Vibration 9
0 05 10minus006
minus004
minus002
0
002
004
006
008
010
z(m
)
t (s)
(a) Vertical position
0 05 10t (s)
minus3
minus2
minus1
0
1
2
3
w(m
middotsminus1)
(b) Transverse speed
0 05 10minus0004
minus0002
0
0002
0004
t (s)
120579(r
ad)
(c) Pitch angle
0 05 10minus10
minus05
0
05
10
t (s)
q(r
ads
)
(d) Pitch rate
0 05 10minus002
0
002
004
006
008
t (s)
hp
(m)
(e) Immersion depth
0 05 10minus60
minus40
minus20
0
20
40
60
80
t (s)
Fp
(N)
(f) Planing force
Figure 12 The motion state of the system when 119896 = 55
When 120590 is equal to 00315 the feedback control laws of120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= minus55119911 are selected which
corresponds to the point (00315 minus55) in the yellow chaoticarea in Figure 2
Figure 11(a) demonstrates that the phase trajectory is achaotic attractor when the control gain of the fin deflectionangle 119896 is equal to minus55 which indicates that the chaoshas occurred and the movement of the vehicle has the
characteristics of a nonlinear dynamic behavior The Lya-punov exponent spectrum corresponding to 119896 = minus55 isgiven in Figure 11(b) It is relatively easy to find that thesystem has a positive Lyapunov exponent and three negativeLyapunov exponents suggesting that the system is in a four-dimensional chaotic state
The motion state of the system is presented in Figure 12After the launch of the supercavitating vehicle the four
10 Shock and Vibration
system state variables namely 119911119908 120579 and 119902 are in the intensenonperiodic oscillating stateThus the vehicle in motion willexperience instability
It can be observed fromFigures 12(e) and 12(f) that withinthe time range of [1 2] (values in 119904) the fin of the vehicleis continuously into contact with the cavity wall under theaction of gravity and produces the planing forceThe planingforce increases gradually with the increase of immersiondepth and the vehicle will rebound inside the cavity whichresults in the loss of the planing force The above actionsrepeat subsequently and the planing force oscillates Theexistence of the planing force will cause vibration and impactto the vehicle resulting in the loss of stability of the vehicleTherefore precise control must be exerted on the vehicle toavoid the above situations [16 17]
5 Conclusions
Thenonlinear dynamic characteristicmovement states underdifferent control parameters of the supercavitating vehiclewere analyzed based on a four-dimensional dynamical modelof the vehicle The following conclusions have been mainlyderived
(1) The movement trajectories of the supercavitatingvehicle have complicated dynamical behavior thesystem will experience Hopf bifurcation periodicalwindows chaos and other nonlinear phenomenawhen the control parameters vary
(2) The movement state of the vehicle under differentcontrol parameters was numerically and preciselyanalyzed according to the phase trajectory diagramthe bifurcation diagram and the Lyapunov exponen-tial spectrum
(3) Most importantly the authors were the first tofind that the range of parameters of the vehicle inany movement state can be determined by a two-dimensional bifurcation diagram The importance ofselecting appropriate control parameters to realize thestable navigation of the supercavitating vehicle wasdemonstrated
It is believed that the work presented in this paper is ofgreat importance for further studies on the stable controlof the underwater supercavitating vehicles especially forengineering practice
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 11472163 11402116 and 61473340)
References
[1] G Wang and M Ostoja-Starzewski ldquoLarge eddy simulationof a sheetcloud cavitation on a NACA0015 hydrofoilrdquo AppliedMathematical Modelling vol 31 no 3 pp 417ndash447 2007
[2] Q T Li Y S He and L P Xue ldquoA numerical simulationof pitching motion of the ventilated supercaviting vehicleapproximately its frontrdquo Chinese Journal of Hydrodynamics vol26 no 6 pp 589ndash685 2011
[3] A K Singhal M M Athavale H Li and Y Jiang ldquoMathemati-cal basis and validation of the full cavitation modelrdquo Journal ofFluids Engineering vol 124 no 3 pp 617ndash624 2002
[4] Y N Savchenko ldquoSupercavitation problems and perspectivesrdquoin Proceedings of the 4th International Symposium onCavitationPasadena Calif USA April 2001
[5] M A Hassouneh and E H Abed ldquoLyapunov and LMI analysisand feedback control of border collision bifurcationsrdquo Nonlin-ear Dynamics vol 50 no 3 pp 373ndash386 2007
[6] Y J Wei J H Wang J Z Zhang W Cao and W H HuangldquoNonlinear dynamics and control of underwater supercavitat-ing vehiclerdquo Journal of Vibration And Shock vol 28 no 6 pp179ndash204 2009
[7] S S Kulkarni and R Pratap ldquoStudies on the dynamics ofa supercavitating projectilerdquo Applied Mathematical Modellingvol 24 no 2 pp 113ndash129 2000
[8] J-Y Choi M Ruzzene and O A Bauchau ldquoDynamic anal-ysis of flexible supercavitating vehicles using modal-basedelementsrdquo Simulation vol 80 no 11 pp 619ndash633 2004
[9] B Feeny ldquoA nonsmooth Coulomb friction oscillatorrdquo Physica DNonlinear Phenomena vol 59 no 1ndash3 pp 25ndash38 1992
[10] A D Vasin and E V Paryshev ldquoImmersion of cylinder in a fluidthrough a cylindrical free surfacerdquo Fluid Dynamics vol 36 no2 pp 169ndash177 2001
[11] G Lin B Balachandran and E Abed ldquoSupercavitating bodydynamics bifurcations and controlrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 691ndash696 PortlandOre USA June 2005
[12] J Dzielski and A Kurdila ldquoA benchmark control problemfor supercavitating vehicles and an initial investigation ofsolutionsrdquo Journal of Vibration amp Control vol 9 no 7 pp 791ndash804 2003
[13] G J Lin B Balakumar and H A Eysd ldquoBifurcation behaviorof a supercavitating vehiclerdquo in Proceedings of the ASMEInternational Mechanical Engineering Congress and Expositionpp 293ndash300 Chicago Ill USA November 2006
[14] G J Lin B Balachandran and E H Abed ldquoDynamics andcontrol of supercavitating vehiclesrdquo Journal of Dynamic SystemsMeasurement and Control vol 130 no 2 Article ID 021003 pp281ndash287 2008
[15] M A Hassouneh V Nguyen B Balachandran and E HAbed ldquoStability analysis and control of supercavitating vehicleswith advection delayrdquo Journal of Computational and NonlinearDynamics vol 8 no 2 Article ID 021003 2012
[16] G Lin B Balachandran and E H Abed ldquoNonlinear dynamicsand bifurcations of a supercavitating vehiclerdquo IEEE Journal ofOceanic Engineering vol 32 no 4 pp 753ndash761 2007
[17] J H Wang Y J Wei and K P Yu ldquoModeling and control ofunderwater supercavitating vehicle based on memory effect ofcavityrdquo Journal of Vibration And Shock vol 29 no 8 pp 160ndash163 2010
the stablemovement of the supercavitating vehicle the effectsof cavitation number 120590 and the fin control law 119896 on thestable movement state of the vehicle were analyzed basedon the four-dimensional dynamical system Here the restof the parameters remain constant and the two-dimensionalbifurcation diagram (120590 119896) is presented in Figure 2 where theparameters are the cavitation number 120590 and the control gain119896 of the fin deflection angle 120575
119890
6 Shock and Vibration
minus06 minus04 minus02 0 02 04
08
10
12
14
16
18
20
q (rads)
w(m
middotsminus1)
(a)
0 5 10 15minus100
minus80
minus60
minus40
minus20
0
20
t (s)
Lyap
unov
expo
nent
spec
trum
(b)
Figure 9 (a) Phase trajectory diagram in 119908 minus 120579 plane when 119896 = 5 and (b) Lyapunov exponent spectrum
In the phase space of (120590 119896) the dynamic behavior of thesystem is presented in Figure 2 The horizontal section isthe bifurcation diagram of the system for different cavitationnumbers 120590 and the vertical section is the bifurcation diagramof the system when the control gain 119896 varies The parameterranges for different system states can be determined bythe two-dimensional bifurcation diagram The region in redrepresents the stable movement state of the vehicle whichmeans the vehicle will navigate steadily when 120590 and 119896 areequal to the values corresponding to any point (120590 119896) withinthis region The green area shows the periodic oscillatorynature of the vehiclemovement whichmeans that the vehiclewill oscillate periodically and hence will become unstableMoreover the vehicle navigating with the states of the yellowarea will suffer from vibration and impact and then collapseWhen the vehicle alters from the steady state to the periodicstate the Hopf bifurcation occurs The boundary betweenthe red and green regions that is the critical switching lineof the stable state and the periodic state is also called theHopf bifurcation line Similarly the boundary between thegreen and yellow areas indicates the switch between theperiodic and chaotic states where the physical phenomenasuch as tangent bifurcation and period doubling bifurcationcan occur
It can be observed from (1) that in the four-dimensionaldynamical system of the underwater vehicle only the planingforce 119865planing is the nonlinear force associated with thesystem state variable and the vertical velocity 119908 This isprimarily attributed to the fact that the complicated nonlinearforce acts on the fin of the vehicle that the vehicle suffersfrom vibration impact and even collapse due to unstablemovement Therefore the nonlinear dynamic characteristicscan be further understood by analyzing the system from thepoint of view of nonlinearity thus preparing for the stablecontrol of the supercavitating vehicle
31 Nonlinear Dynamical Characteristic of the Vehicle underDifferent Cavitation Values According to the dynamicalbehavior distribution diagram presented in Figure 2 thebifurcation diagram between the system state variable 119908 and
the cavitation number 120590 is provided in Figure 3 (119896 = 1 ie120575119890= 119911 and 120575
119888= 15119911 minus 30120579 minus 03119902) Some simple explanations
are given as followsWhen the cavitation number 120590 of the system falls in
the range of [00198 002687] the trajectory of the vehicleconverges to a stable equilibrium point
When 120590 is equal to 002687 the Hopf bifurcation occursas a result of which the stable equilibrium point becomes thestable periodic trajectory and the vehicle oscillates periodi-cally
After a series of period doubling bifurcation the systemfalls into a chaotic state and the vehicle suffers from hugeimpact When the cavitation number 120590 is approximately003083 the system has three stable periodic trajectories
The bifurcation diagram shown in Figure 3 when 120590 isin
[00315 00325] is magnified in Figure 4 which depicts thediversified bifurcation behaviors of the system
After a series of period doubling bifurcations the systemshifts from three periodic trajectories into three huge chaoticattractors respectively when 120590 is approximately 003197 thisphenomenon is referred to as chaos crisis [12]
When 120590 is equal to 0032 the chaotic attractors suddenlychange into periodic trajectories and form one period-2window and two period-3 windows This phenomenon isreferred to as tangent bifurcation The tangent bifurcationwill cause intermittent chaos and the periodic trajectoriessuddenly develop chaotic bands in the periodic window afterexperiencing a period doubling bifurcation
When 120590 is equal to 003204 the secondary chaotic bandcoincides with the unstable periodic trajectories which thencauses the chaotic crisis The secondary narrow chaotic bandwill then transform into a broad chaotic band
With the increase of 120590 the obvious period-2 windowoccurs for 120590 isin [003207 003225] and when 120590 is approxi-mately 0032228 the broad chaotic band suddenly changesinto two periodic trajectories
32 Nonlinear Dynamic Characteristic of the Vehicle underDifferent Fin Deflection Angles When the cavitation number
Shock and Vibration 7
10 15 200044
0045
0046
0047
0048
z(m
)
t (s)
(a) Vertical position
10 15 2010
15
20
t (s)
w(m
middotsminus1)
(b) Transverse speed
10 15 200020
0021
0022
t (s)
120579(r
ad)
(c) Pitch angle
10 15 20minus03
minus02
minus01
0
01
02
03
q(r
ads
)
t (s)
(d) Pitch rate
minus002
0
002
004
006
008
15 2010t (s)
hp
(m)
(e) Immersion depth
10 15 20minus20
0
20
40
60
80
t (s)
Fp
(N)
(f) Planing force
Figure 10 The motion state of the system when 119896 = 5
is set as 120590 = 00315 the cavitator deflection angle is 120575119888=
15119911 minus 30120579 minus 03119902 and the fin deflection angle is 120575119890= 119896119911 The
bifurcation diagram of the supercavitating vehicle betweenthe system state variable 119908 and the control gain 119896 of the findeflection angle is presented in Figure 5 According to thedynamical behavior distribution presented in Figure 2 the
effective range of the control gain 119896 for the supercavitatingvehicle is presented in Figure 5 when 120590 = 00315
Figure 5 shows that when 119896 falls within the wider range[minus7614 minus5238] the system is in a chaotic state and finallythe period-2 trajectory occurs through period doublingbifurcation
8 Shock and Vibration
minus04 minus02 0 02 04 06minus20
minus10
0
10
20
q (rads)
w(m
middotsminus1)
(a)
0 5 10 15
minus100
minus80
minus60
minus40
minus20
0
20
Lyap
unov
expo
nent
spec
trum
t (s)
(b)
Figure 11 (a) Phase trajectory diagram in 119908 minus 120579 plane when 119896 = minus55 and (b) Lyapunov exponent spectrum
When 119896 is approximately 856 the periodic state endsand the system is in the divergent state The correspondingmagnified part in Figure 5 for minus80 lt 119896 lt minus50 is givenin Figure 6 which indicates that the system changes from aperiodic state into a chaotic state when 119896 = minus779
When 119896 is approximately minus7754 the tangent bifurcationoccurs leading to an intermittent chaos and forming theperiod-3 windows and then three stable periodic trajectories
The period doubling bifurcations occur for the threetrajectories when 119896 is approximately minus77 minus7616 or minus755When 119896 is approximately minus746 the secondary chaotic bandand the unstable periodic trajectories converge into chaos
When 119896 is approximately minus7358 the tangent bifurcationoccurs The system suddenly switches from chaotic state toperiodic state and the period-three windows forms With theoccurrence of a series of period doubling bifurcation thesystem enters into the chaotic state again when 119896 is betweenminus7118 andminus5264 uponwhich the system switches back fromthe chaotic state to the periodic state
4 Movement Characteristic Analysis of theUnderwater Supercavitating Vehicle
When the system of the supercavitating vehicle is not con-trolled the movement state of the system is unstable [12 14]To investigate its movement characteristics alone accordingto the two-dimensional bifurcation diagram the rest of theparameters of the system should be kept constant Assumingthat 120590 is equal to 00315 the feedback control laws 120575
119888=
15119911 minus 30120579 minus 03119902 and 120575119890= minus2195119911 which corresponds to
the point (00315 minus2195) in the red stable movement area inFigure 2Thephase trajectory diagram is shown in Figure 7(a)when the control gain of the fin deflection angle 119896 is equal tominus2195
It can be observed that when 119896 = minus2195 the phasetrajectory of the system gradually stabilizes at an equilibriumpointThe Lyapunov exponent spectrum as a function of time
is presented in Figure 7(b) in which the values of the largestLyapunov exponent curve are negative within a finite time
The motion state of the supercavitating vehicle is pre-sented in Figure 8 inwhich the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate 119902 are attracted to the equilibrium point(00047 00866 00012 0) with less settling time under thecontrol of the law of stable movement
Figures 8(e) and 8(f) demonstrate that the immersiondepth ℎ
1015840 of the fin and the corresponding planing force 119865119901
are both 0 This indicates that the fin is inside the cavity anddoes not have any contact with the cavity the vehicle is in astable navigation state
When 120590 is equal to 00315 the feedback control laws are120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= 3119911 which corresponds to the
point (00315 3) in the green periodic oscillation region inFigure 2
Figure 9(a) tells us that the phase trajectory is a limitcycle with period 2 when the control gain of the fin deflectionangle 119896 is equal to 3 The Lyapunov exponent spectrumcorresponding to 119896 = 3 is shown in Figure 9(b) It is notdifficult to find that the system approximately has a zeroLyapunov exponent and three negative Lyapunov exponents
The motion state of the supercavitating vehicle is shownin Figure 10 in which the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate velocity 119902 oscillate periodically at theequilibrium point (00416 13937 00190 0)
The immersion depth ℎ1015840 of the fin oscillates periodically
in the range of [0 006] (in m) which indicates that thevehicle continuously collides with the cavity wall
The fin is inside the cavity at times and does not come incontact with the cavity which results in zero planing force119865119901 The fin penetrates the cavity into the water at times
and produces the planing force oscillating periodically in therange of [0 48] (in N) The above actions repeat again andagain and such phenomenon is referred to as ldquofin attackphenomenonrdquo It also indicates that the vehicle is in anunstable periodic oscillating state
Shock and Vibration 9
0 05 10minus006
minus004
minus002
0
002
004
006
008
010
z(m
)
t (s)
(a) Vertical position
0 05 10t (s)
minus3
minus2
minus1
0
1
2
3
w(m
middotsminus1)
(b) Transverse speed
0 05 10minus0004
minus0002
0
0002
0004
t (s)
120579(r
ad)
(c) Pitch angle
0 05 10minus10
minus05
0
05
10
t (s)
q(r
ads
)
(d) Pitch rate
0 05 10minus002
0
002
004
006
008
t (s)
hp
(m)
(e) Immersion depth
0 05 10minus60
minus40
minus20
0
20
40
60
80
t (s)
Fp
(N)
(f) Planing force
Figure 12 The motion state of the system when 119896 = 55
When 120590 is equal to 00315 the feedback control laws of120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= minus55119911 are selected which
corresponds to the point (00315 minus55) in the yellow chaoticarea in Figure 2
Figure 11(a) demonstrates that the phase trajectory is achaotic attractor when the control gain of the fin deflectionangle 119896 is equal to minus55 which indicates that the chaoshas occurred and the movement of the vehicle has the
characteristics of a nonlinear dynamic behavior The Lya-punov exponent spectrum corresponding to 119896 = minus55 isgiven in Figure 11(b) It is relatively easy to find that thesystem has a positive Lyapunov exponent and three negativeLyapunov exponents suggesting that the system is in a four-dimensional chaotic state
The motion state of the system is presented in Figure 12After the launch of the supercavitating vehicle the four
10 Shock and Vibration
system state variables namely 119911119908 120579 and 119902 are in the intensenonperiodic oscillating stateThus the vehicle in motion willexperience instability
It can be observed fromFigures 12(e) and 12(f) that withinthe time range of [1 2] (values in 119904) the fin of the vehicleis continuously into contact with the cavity wall under theaction of gravity and produces the planing forceThe planingforce increases gradually with the increase of immersiondepth and the vehicle will rebound inside the cavity whichresults in the loss of the planing force The above actionsrepeat subsequently and the planing force oscillates Theexistence of the planing force will cause vibration and impactto the vehicle resulting in the loss of stability of the vehicleTherefore precise control must be exerted on the vehicle toavoid the above situations [16 17]
5 Conclusions
Thenonlinear dynamic characteristicmovement states underdifferent control parameters of the supercavitating vehiclewere analyzed based on a four-dimensional dynamical modelof the vehicle The following conclusions have been mainlyderived
(1) The movement trajectories of the supercavitatingvehicle have complicated dynamical behavior thesystem will experience Hopf bifurcation periodicalwindows chaos and other nonlinear phenomenawhen the control parameters vary
(2) The movement state of the vehicle under differentcontrol parameters was numerically and preciselyanalyzed according to the phase trajectory diagramthe bifurcation diagram and the Lyapunov exponen-tial spectrum
(3) Most importantly the authors were the first tofind that the range of parameters of the vehicle inany movement state can be determined by a two-dimensional bifurcation diagram The importance ofselecting appropriate control parameters to realize thestable navigation of the supercavitating vehicle wasdemonstrated
It is believed that the work presented in this paper is ofgreat importance for further studies on the stable controlof the underwater supercavitating vehicles especially forengineering practice
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 11472163 11402116 and 61473340)
References
[1] G Wang and M Ostoja-Starzewski ldquoLarge eddy simulationof a sheetcloud cavitation on a NACA0015 hydrofoilrdquo AppliedMathematical Modelling vol 31 no 3 pp 417ndash447 2007
[2] Q T Li Y S He and L P Xue ldquoA numerical simulationof pitching motion of the ventilated supercaviting vehicleapproximately its frontrdquo Chinese Journal of Hydrodynamics vol26 no 6 pp 589ndash685 2011
[3] A K Singhal M M Athavale H Li and Y Jiang ldquoMathemati-cal basis and validation of the full cavitation modelrdquo Journal ofFluids Engineering vol 124 no 3 pp 617ndash624 2002
[4] Y N Savchenko ldquoSupercavitation problems and perspectivesrdquoin Proceedings of the 4th International Symposium onCavitationPasadena Calif USA April 2001
[5] M A Hassouneh and E H Abed ldquoLyapunov and LMI analysisand feedback control of border collision bifurcationsrdquo Nonlin-ear Dynamics vol 50 no 3 pp 373ndash386 2007
[6] Y J Wei J H Wang J Z Zhang W Cao and W H HuangldquoNonlinear dynamics and control of underwater supercavitat-ing vehiclerdquo Journal of Vibration And Shock vol 28 no 6 pp179ndash204 2009
[7] S S Kulkarni and R Pratap ldquoStudies on the dynamics ofa supercavitating projectilerdquo Applied Mathematical Modellingvol 24 no 2 pp 113ndash129 2000
[8] J-Y Choi M Ruzzene and O A Bauchau ldquoDynamic anal-ysis of flexible supercavitating vehicles using modal-basedelementsrdquo Simulation vol 80 no 11 pp 619ndash633 2004
[9] B Feeny ldquoA nonsmooth Coulomb friction oscillatorrdquo Physica DNonlinear Phenomena vol 59 no 1ndash3 pp 25ndash38 1992
[10] A D Vasin and E V Paryshev ldquoImmersion of cylinder in a fluidthrough a cylindrical free surfacerdquo Fluid Dynamics vol 36 no2 pp 169ndash177 2001
[11] G Lin B Balachandran and E Abed ldquoSupercavitating bodydynamics bifurcations and controlrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 691ndash696 PortlandOre USA June 2005
[12] J Dzielski and A Kurdila ldquoA benchmark control problemfor supercavitating vehicles and an initial investigation ofsolutionsrdquo Journal of Vibration amp Control vol 9 no 7 pp 791ndash804 2003
[13] G J Lin B Balakumar and H A Eysd ldquoBifurcation behaviorof a supercavitating vehiclerdquo in Proceedings of the ASMEInternational Mechanical Engineering Congress and Expositionpp 293ndash300 Chicago Ill USA November 2006
[14] G J Lin B Balachandran and E H Abed ldquoDynamics andcontrol of supercavitating vehiclesrdquo Journal of Dynamic SystemsMeasurement and Control vol 130 no 2 Article ID 021003 pp281ndash287 2008
[15] M A Hassouneh V Nguyen B Balachandran and E HAbed ldquoStability analysis and control of supercavitating vehicleswith advection delayrdquo Journal of Computational and NonlinearDynamics vol 8 no 2 Article ID 021003 2012
[16] G Lin B Balachandran and E H Abed ldquoNonlinear dynamicsand bifurcations of a supercavitating vehiclerdquo IEEE Journal ofOceanic Engineering vol 32 no 4 pp 753ndash761 2007
[17] J H Wang Y J Wei and K P Yu ldquoModeling and control ofunderwater supercavitating vehicle based on memory effect ofcavityrdquo Journal of Vibration And Shock vol 29 no 8 pp 160ndash163 2010
the stablemovement of the supercavitating vehicle the effectsof cavitation number 120590 and the fin control law 119896 on thestable movement state of the vehicle were analyzed basedon the four-dimensional dynamical system Here the restof the parameters remain constant and the two-dimensionalbifurcation diagram (120590 119896) is presented in Figure 2 where theparameters are the cavitation number 120590 and the control gain119896 of the fin deflection angle 120575
119890
6 Shock and Vibration
minus06 minus04 minus02 0 02 04
08
10
12
14
16
18
20
q (rads)
w(m
middotsminus1)
(a)
0 5 10 15minus100
minus80
minus60
minus40
minus20
0
20
t (s)
Lyap
unov
expo
nent
spec
trum
(b)
Figure 9 (a) Phase trajectory diagram in 119908 minus 120579 plane when 119896 = 5 and (b) Lyapunov exponent spectrum
In the phase space of (120590 119896) the dynamic behavior of thesystem is presented in Figure 2 The horizontal section isthe bifurcation diagram of the system for different cavitationnumbers 120590 and the vertical section is the bifurcation diagramof the system when the control gain 119896 varies The parameterranges for different system states can be determined bythe two-dimensional bifurcation diagram The region in redrepresents the stable movement state of the vehicle whichmeans the vehicle will navigate steadily when 120590 and 119896 areequal to the values corresponding to any point (120590 119896) withinthis region The green area shows the periodic oscillatorynature of the vehiclemovement whichmeans that the vehiclewill oscillate periodically and hence will become unstableMoreover the vehicle navigating with the states of the yellowarea will suffer from vibration and impact and then collapseWhen the vehicle alters from the steady state to the periodicstate the Hopf bifurcation occurs The boundary betweenthe red and green regions that is the critical switching lineof the stable state and the periodic state is also called theHopf bifurcation line Similarly the boundary between thegreen and yellow areas indicates the switch between theperiodic and chaotic states where the physical phenomenasuch as tangent bifurcation and period doubling bifurcationcan occur
It can be observed from (1) that in the four-dimensionaldynamical system of the underwater vehicle only the planingforce 119865planing is the nonlinear force associated with thesystem state variable and the vertical velocity 119908 This isprimarily attributed to the fact that the complicated nonlinearforce acts on the fin of the vehicle that the vehicle suffersfrom vibration impact and even collapse due to unstablemovement Therefore the nonlinear dynamic characteristicscan be further understood by analyzing the system from thepoint of view of nonlinearity thus preparing for the stablecontrol of the supercavitating vehicle
31 Nonlinear Dynamical Characteristic of the Vehicle underDifferent Cavitation Values According to the dynamicalbehavior distribution diagram presented in Figure 2 thebifurcation diagram between the system state variable 119908 and
the cavitation number 120590 is provided in Figure 3 (119896 = 1 ie120575119890= 119911 and 120575
119888= 15119911 minus 30120579 minus 03119902) Some simple explanations
are given as followsWhen the cavitation number 120590 of the system falls in
the range of [00198 002687] the trajectory of the vehicleconverges to a stable equilibrium point
When 120590 is equal to 002687 the Hopf bifurcation occursas a result of which the stable equilibrium point becomes thestable periodic trajectory and the vehicle oscillates periodi-cally
After a series of period doubling bifurcation the systemfalls into a chaotic state and the vehicle suffers from hugeimpact When the cavitation number 120590 is approximately003083 the system has three stable periodic trajectories
The bifurcation diagram shown in Figure 3 when 120590 isin
[00315 00325] is magnified in Figure 4 which depicts thediversified bifurcation behaviors of the system
After a series of period doubling bifurcations the systemshifts from three periodic trajectories into three huge chaoticattractors respectively when 120590 is approximately 003197 thisphenomenon is referred to as chaos crisis [12]
When 120590 is equal to 0032 the chaotic attractors suddenlychange into periodic trajectories and form one period-2window and two period-3 windows This phenomenon isreferred to as tangent bifurcation The tangent bifurcationwill cause intermittent chaos and the periodic trajectoriessuddenly develop chaotic bands in the periodic window afterexperiencing a period doubling bifurcation
When 120590 is equal to 003204 the secondary chaotic bandcoincides with the unstable periodic trajectories which thencauses the chaotic crisis The secondary narrow chaotic bandwill then transform into a broad chaotic band
With the increase of 120590 the obvious period-2 windowoccurs for 120590 isin [003207 003225] and when 120590 is approxi-mately 0032228 the broad chaotic band suddenly changesinto two periodic trajectories
32 Nonlinear Dynamic Characteristic of the Vehicle underDifferent Fin Deflection Angles When the cavitation number
Shock and Vibration 7
10 15 200044
0045
0046
0047
0048
z(m
)
t (s)
(a) Vertical position
10 15 2010
15
20
t (s)
w(m
middotsminus1)
(b) Transverse speed
10 15 200020
0021
0022
t (s)
120579(r
ad)
(c) Pitch angle
10 15 20minus03
minus02
minus01
0
01
02
03
q(r
ads
)
t (s)
(d) Pitch rate
minus002
0
002
004
006
008
15 2010t (s)
hp
(m)
(e) Immersion depth
10 15 20minus20
0
20
40
60
80
t (s)
Fp
(N)
(f) Planing force
Figure 10 The motion state of the system when 119896 = 5
is set as 120590 = 00315 the cavitator deflection angle is 120575119888=
15119911 minus 30120579 minus 03119902 and the fin deflection angle is 120575119890= 119896119911 The
bifurcation diagram of the supercavitating vehicle betweenthe system state variable 119908 and the control gain 119896 of the findeflection angle is presented in Figure 5 According to thedynamical behavior distribution presented in Figure 2 the
effective range of the control gain 119896 for the supercavitatingvehicle is presented in Figure 5 when 120590 = 00315
Figure 5 shows that when 119896 falls within the wider range[minus7614 minus5238] the system is in a chaotic state and finallythe period-2 trajectory occurs through period doublingbifurcation
8 Shock and Vibration
minus04 minus02 0 02 04 06minus20
minus10
0
10
20
q (rads)
w(m
middotsminus1)
(a)
0 5 10 15
minus100
minus80
minus60
minus40
minus20
0
20
Lyap
unov
expo
nent
spec
trum
t (s)
(b)
Figure 11 (a) Phase trajectory diagram in 119908 minus 120579 plane when 119896 = minus55 and (b) Lyapunov exponent spectrum
When 119896 is approximately 856 the periodic state endsand the system is in the divergent state The correspondingmagnified part in Figure 5 for minus80 lt 119896 lt minus50 is givenin Figure 6 which indicates that the system changes from aperiodic state into a chaotic state when 119896 = minus779
When 119896 is approximately minus7754 the tangent bifurcationoccurs leading to an intermittent chaos and forming theperiod-3 windows and then three stable periodic trajectories
The period doubling bifurcations occur for the threetrajectories when 119896 is approximately minus77 minus7616 or minus755When 119896 is approximately minus746 the secondary chaotic bandand the unstable periodic trajectories converge into chaos
When 119896 is approximately minus7358 the tangent bifurcationoccurs The system suddenly switches from chaotic state toperiodic state and the period-three windows forms With theoccurrence of a series of period doubling bifurcation thesystem enters into the chaotic state again when 119896 is betweenminus7118 andminus5264 uponwhich the system switches back fromthe chaotic state to the periodic state
4 Movement Characteristic Analysis of theUnderwater Supercavitating Vehicle
When the system of the supercavitating vehicle is not con-trolled the movement state of the system is unstable [12 14]To investigate its movement characteristics alone accordingto the two-dimensional bifurcation diagram the rest of theparameters of the system should be kept constant Assumingthat 120590 is equal to 00315 the feedback control laws 120575
119888=
15119911 minus 30120579 minus 03119902 and 120575119890= minus2195119911 which corresponds to
the point (00315 minus2195) in the red stable movement area inFigure 2Thephase trajectory diagram is shown in Figure 7(a)when the control gain of the fin deflection angle 119896 is equal tominus2195
It can be observed that when 119896 = minus2195 the phasetrajectory of the system gradually stabilizes at an equilibriumpointThe Lyapunov exponent spectrum as a function of time
is presented in Figure 7(b) in which the values of the largestLyapunov exponent curve are negative within a finite time
The motion state of the supercavitating vehicle is pre-sented in Figure 8 inwhich the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate 119902 are attracted to the equilibrium point(00047 00866 00012 0) with less settling time under thecontrol of the law of stable movement
Figures 8(e) and 8(f) demonstrate that the immersiondepth ℎ
1015840 of the fin and the corresponding planing force 119865119901
are both 0 This indicates that the fin is inside the cavity anddoes not have any contact with the cavity the vehicle is in astable navigation state
When 120590 is equal to 00315 the feedback control laws are120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= 3119911 which corresponds to the
point (00315 3) in the green periodic oscillation region inFigure 2
Figure 9(a) tells us that the phase trajectory is a limitcycle with period 2 when the control gain of the fin deflectionangle 119896 is equal to 3 The Lyapunov exponent spectrumcorresponding to 119896 = 3 is shown in Figure 9(b) It is notdifficult to find that the system approximately has a zeroLyapunov exponent and three negative Lyapunov exponents
The motion state of the supercavitating vehicle is shownin Figure 10 in which the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate velocity 119902 oscillate periodically at theequilibrium point (00416 13937 00190 0)
The immersion depth ℎ1015840 of the fin oscillates periodically
in the range of [0 006] (in m) which indicates that thevehicle continuously collides with the cavity wall
The fin is inside the cavity at times and does not come incontact with the cavity which results in zero planing force119865119901 The fin penetrates the cavity into the water at times
and produces the planing force oscillating periodically in therange of [0 48] (in N) The above actions repeat again andagain and such phenomenon is referred to as ldquofin attackphenomenonrdquo It also indicates that the vehicle is in anunstable periodic oscillating state
Shock and Vibration 9
0 05 10minus006
minus004
minus002
0
002
004
006
008
010
z(m
)
t (s)
(a) Vertical position
0 05 10t (s)
minus3
minus2
minus1
0
1
2
3
w(m
middotsminus1)
(b) Transverse speed
0 05 10minus0004
minus0002
0
0002
0004
t (s)
120579(r
ad)
(c) Pitch angle
0 05 10minus10
minus05
0
05
10
t (s)
q(r
ads
)
(d) Pitch rate
0 05 10minus002
0
002
004
006
008
t (s)
hp
(m)
(e) Immersion depth
0 05 10minus60
minus40
minus20
0
20
40
60
80
t (s)
Fp
(N)
(f) Planing force
Figure 12 The motion state of the system when 119896 = 55
When 120590 is equal to 00315 the feedback control laws of120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= minus55119911 are selected which
corresponds to the point (00315 minus55) in the yellow chaoticarea in Figure 2
Figure 11(a) demonstrates that the phase trajectory is achaotic attractor when the control gain of the fin deflectionangle 119896 is equal to minus55 which indicates that the chaoshas occurred and the movement of the vehicle has the
characteristics of a nonlinear dynamic behavior The Lya-punov exponent spectrum corresponding to 119896 = minus55 isgiven in Figure 11(b) It is relatively easy to find that thesystem has a positive Lyapunov exponent and three negativeLyapunov exponents suggesting that the system is in a four-dimensional chaotic state
The motion state of the system is presented in Figure 12After the launch of the supercavitating vehicle the four
10 Shock and Vibration
system state variables namely 119911119908 120579 and 119902 are in the intensenonperiodic oscillating stateThus the vehicle in motion willexperience instability
It can be observed fromFigures 12(e) and 12(f) that withinthe time range of [1 2] (values in 119904) the fin of the vehicleis continuously into contact with the cavity wall under theaction of gravity and produces the planing forceThe planingforce increases gradually with the increase of immersiondepth and the vehicle will rebound inside the cavity whichresults in the loss of the planing force The above actionsrepeat subsequently and the planing force oscillates Theexistence of the planing force will cause vibration and impactto the vehicle resulting in the loss of stability of the vehicleTherefore precise control must be exerted on the vehicle toavoid the above situations [16 17]
5 Conclusions
Thenonlinear dynamic characteristicmovement states underdifferent control parameters of the supercavitating vehiclewere analyzed based on a four-dimensional dynamical modelof the vehicle The following conclusions have been mainlyderived
(1) The movement trajectories of the supercavitatingvehicle have complicated dynamical behavior thesystem will experience Hopf bifurcation periodicalwindows chaos and other nonlinear phenomenawhen the control parameters vary
(2) The movement state of the vehicle under differentcontrol parameters was numerically and preciselyanalyzed according to the phase trajectory diagramthe bifurcation diagram and the Lyapunov exponen-tial spectrum
(3) Most importantly the authors were the first tofind that the range of parameters of the vehicle inany movement state can be determined by a two-dimensional bifurcation diagram The importance ofselecting appropriate control parameters to realize thestable navigation of the supercavitating vehicle wasdemonstrated
It is believed that the work presented in this paper is ofgreat importance for further studies on the stable controlof the underwater supercavitating vehicles especially forengineering practice
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 11472163 11402116 and 61473340)
References
[1] G Wang and M Ostoja-Starzewski ldquoLarge eddy simulationof a sheetcloud cavitation on a NACA0015 hydrofoilrdquo AppliedMathematical Modelling vol 31 no 3 pp 417ndash447 2007
[2] Q T Li Y S He and L P Xue ldquoA numerical simulationof pitching motion of the ventilated supercaviting vehicleapproximately its frontrdquo Chinese Journal of Hydrodynamics vol26 no 6 pp 589ndash685 2011
[3] A K Singhal M M Athavale H Li and Y Jiang ldquoMathemati-cal basis and validation of the full cavitation modelrdquo Journal ofFluids Engineering vol 124 no 3 pp 617ndash624 2002
[4] Y N Savchenko ldquoSupercavitation problems and perspectivesrdquoin Proceedings of the 4th International Symposium onCavitationPasadena Calif USA April 2001
[5] M A Hassouneh and E H Abed ldquoLyapunov and LMI analysisand feedback control of border collision bifurcationsrdquo Nonlin-ear Dynamics vol 50 no 3 pp 373ndash386 2007
[6] Y J Wei J H Wang J Z Zhang W Cao and W H HuangldquoNonlinear dynamics and control of underwater supercavitat-ing vehiclerdquo Journal of Vibration And Shock vol 28 no 6 pp179ndash204 2009
[7] S S Kulkarni and R Pratap ldquoStudies on the dynamics ofa supercavitating projectilerdquo Applied Mathematical Modellingvol 24 no 2 pp 113ndash129 2000
[8] J-Y Choi M Ruzzene and O A Bauchau ldquoDynamic anal-ysis of flexible supercavitating vehicles using modal-basedelementsrdquo Simulation vol 80 no 11 pp 619ndash633 2004
[9] B Feeny ldquoA nonsmooth Coulomb friction oscillatorrdquo Physica DNonlinear Phenomena vol 59 no 1ndash3 pp 25ndash38 1992
[10] A D Vasin and E V Paryshev ldquoImmersion of cylinder in a fluidthrough a cylindrical free surfacerdquo Fluid Dynamics vol 36 no2 pp 169ndash177 2001
[11] G Lin B Balachandran and E Abed ldquoSupercavitating bodydynamics bifurcations and controlrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 691ndash696 PortlandOre USA June 2005
[12] J Dzielski and A Kurdila ldquoA benchmark control problemfor supercavitating vehicles and an initial investigation ofsolutionsrdquo Journal of Vibration amp Control vol 9 no 7 pp 791ndash804 2003
[13] G J Lin B Balakumar and H A Eysd ldquoBifurcation behaviorof a supercavitating vehiclerdquo in Proceedings of the ASMEInternational Mechanical Engineering Congress and Expositionpp 293ndash300 Chicago Ill USA November 2006
[14] G J Lin B Balachandran and E H Abed ldquoDynamics andcontrol of supercavitating vehiclesrdquo Journal of Dynamic SystemsMeasurement and Control vol 130 no 2 Article ID 021003 pp281ndash287 2008
[15] M A Hassouneh V Nguyen B Balachandran and E HAbed ldquoStability analysis and control of supercavitating vehicleswith advection delayrdquo Journal of Computational and NonlinearDynamics vol 8 no 2 Article ID 021003 2012
[16] G Lin B Balachandran and E H Abed ldquoNonlinear dynamicsand bifurcations of a supercavitating vehiclerdquo IEEE Journal ofOceanic Engineering vol 32 no 4 pp 753ndash761 2007
[17] J H Wang Y J Wei and K P Yu ldquoModeling and control ofunderwater supercavitating vehicle based on memory effect ofcavityrdquo Journal of Vibration And Shock vol 29 no 8 pp 160ndash163 2010
Figure 9 (a) Phase trajectory diagram in 119908 minus 120579 plane when 119896 = 5 and (b) Lyapunov exponent spectrum
In the phase space of (120590 119896) the dynamic behavior of thesystem is presented in Figure 2 The horizontal section isthe bifurcation diagram of the system for different cavitationnumbers 120590 and the vertical section is the bifurcation diagramof the system when the control gain 119896 varies The parameterranges for different system states can be determined bythe two-dimensional bifurcation diagram The region in redrepresents the stable movement state of the vehicle whichmeans the vehicle will navigate steadily when 120590 and 119896 areequal to the values corresponding to any point (120590 119896) withinthis region The green area shows the periodic oscillatorynature of the vehiclemovement whichmeans that the vehiclewill oscillate periodically and hence will become unstableMoreover the vehicle navigating with the states of the yellowarea will suffer from vibration and impact and then collapseWhen the vehicle alters from the steady state to the periodicstate the Hopf bifurcation occurs The boundary betweenthe red and green regions that is the critical switching lineof the stable state and the periodic state is also called theHopf bifurcation line Similarly the boundary between thegreen and yellow areas indicates the switch between theperiodic and chaotic states where the physical phenomenasuch as tangent bifurcation and period doubling bifurcationcan occur
It can be observed from (1) that in the four-dimensionaldynamical system of the underwater vehicle only the planingforce 119865planing is the nonlinear force associated with thesystem state variable and the vertical velocity 119908 This isprimarily attributed to the fact that the complicated nonlinearforce acts on the fin of the vehicle that the vehicle suffersfrom vibration impact and even collapse due to unstablemovement Therefore the nonlinear dynamic characteristicscan be further understood by analyzing the system from thepoint of view of nonlinearity thus preparing for the stablecontrol of the supercavitating vehicle
31 Nonlinear Dynamical Characteristic of the Vehicle underDifferent Cavitation Values According to the dynamicalbehavior distribution diagram presented in Figure 2 thebifurcation diagram between the system state variable 119908 and
the cavitation number 120590 is provided in Figure 3 (119896 = 1 ie120575119890= 119911 and 120575
119888= 15119911 minus 30120579 minus 03119902) Some simple explanations
are given as followsWhen the cavitation number 120590 of the system falls in
the range of [00198 002687] the trajectory of the vehicleconverges to a stable equilibrium point
When 120590 is equal to 002687 the Hopf bifurcation occursas a result of which the stable equilibrium point becomes thestable periodic trajectory and the vehicle oscillates periodi-cally
After a series of period doubling bifurcation the systemfalls into a chaotic state and the vehicle suffers from hugeimpact When the cavitation number 120590 is approximately003083 the system has three stable periodic trajectories
The bifurcation diagram shown in Figure 3 when 120590 isin
[00315 00325] is magnified in Figure 4 which depicts thediversified bifurcation behaviors of the system
After a series of period doubling bifurcations the systemshifts from three periodic trajectories into three huge chaoticattractors respectively when 120590 is approximately 003197 thisphenomenon is referred to as chaos crisis [12]
When 120590 is equal to 0032 the chaotic attractors suddenlychange into periodic trajectories and form one period-2window and two period-3 windows This phenomenon isreferred to as tangent bifurcation The tangent bifurcationwill cause intermittent chaos and the periodic trajectoriessuddenly develop chaotic bands in the periodic window afterexperiencing a period doubling bifurcation
When 120590 is equal to 003204 the secondary chaotic bandcoincides with the unstable periodic trajectories which thencauses the chaotic crisis The secondary narrow chaotic bandwill then transform into a broad chaotic band
With the increase of 120590 the obvious period-2 windowoccurs for 120590 isin [003207 003225] and when 120590 is approxi-mately 0032228 the broad chaotic band suddenly changesinto two periodic trajectories
32 Nonlinear Dynamic Characteristic of the Vehicle underDifferent Fin Deflection Angles When the cavitation number
Shock and Vibration 7
10 15 200044
0045
0046
0047
0048
z(m
)
t (s)
(a) Vertical position
10 15 2010
15
20
t (s)
w(m
middotsminus1)
(b) Transverse speed
10 15 200020
0021
0022
t (s)
120579(r
ad)
(c) Pitch angle
10 15 20minus03
minus02
minus01
0
01
02
03
q(r
ads
)
t (s)
(d) Pitch rate
minus002
0
002
004
006
008
15 2010t (s)
hp
(m)
(e) Immersion depth
10 15 20minus20
0
20
40
60
80
t (s)
Fp
(N)
(f) Planing force
Figure 10 The motion state of the system when 119896 = 5
is set as 120590 = 00315 the cavitator deflection angle is 120575119888=
15119911 minus 30120579 minus 03119902 and the fin deflection angle is 120575119890= 119896119911 The
bifurcation diagram of the supercavitating vehicle betweenthe system state variable 119908 and the control gain 119896 of the findeflection angle is presented in Figure 5 According to thedynamical behavior distribution presented in Figure 2 the
effective range of the control gain 119896 for the supercavitatingvehicle is presented in Figure 5 when 120590 = 00315
Figure 5 shows that when 119896 falls within the wider range[minus7614 minus5238] the system is in a chaotic state and finallythe period-2 trajectory occurs through period doublingbifurcation
8 Shock and Vibration
minus04 minus02 0 02 04 06minus20
minus10
0
10
20
q (rads)
w(m
middotsminus1)
(a)
0 5 10 15
minus100
minus80
minus60
minus40
minus20
0
20
Lyap
unov
expo
nent
spec
trum
t (s)
(b)
Figure 11 (a) Phase trajectory diagram in 119908 minus 120579 plane when 119896 = minus55 and (b) Lyapunov exponent spectrum
When 119896 is approximately 856 the periodic state endsand the system is in the divergent state The correspondingmagnified part in Figure 5 for minus80 lt 119896 lt minus50 is givenin Figure 6 which indicates that the system changes from aperiodic state into a chaotic state when 119896 = minus779
When 119896 is approximately minus7754 the tangent bifurcationoccurs leading to an intermittent chaos and forming theperiod-3 windows and then three stable periodic trajectories
The period doubling bifurcations occur for the threetrajectories when 119896 is approximately minus77 minus7616 or minus755When 119896 is approximately minus746 the secondary chaotic bandand the unstable periodic trajectories converge into chaos
When 119896 is approximately minus7358 the tangent bifurcationoccurs The system suddenly switches from chaotic state toperiodic state and the period-three windows forms With theoccurrence of a series of period doubling bifurcation thesystem enters into the chaotic state again when 119896 is betweenminus7118 andminus5264 uponwhich the system switches back fromthe chaotic state to the periodic state
4 Movement Characteristic Analysis of theUnderwater Supercavitating Vehicle
When the system of the supercavitating vehicle is not con-trolled the movement state of the system is unstable [12 14]To investigate its movement characteristics alone accordingto the two-dimensional bifurcation diagram the rest of theparameters of the system should be kept constant Assumingthat 120590 is equal to 00315 the feedback control laws 120575
119888=
15119911 minus 30120579 minus 03119902 and 120575119890= minus2195119911 which corresponds to
the point (00315 minus2195) in the red stable movement area inFigure 2Thephase trajectory diagram is shown in Figure 7(a)when the control gain of the fin deflection angle 119896 is equal tominus2195
It can be observed that when 119896 = minus2195 the phasetrajectory of the system gradually stabilizes at an equilibriumpointThe Lyapunov exponent spectrum as a function of time
is presented in Figure 7(b) in which the values of the largestLyapunov exponent curve are negative within a finite time
The motion state of the supercavitating vehicle is pre-sented in Figure 8 inwhich the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate 119902 are attracted to the equilibrium point(00047 00866 00012 0) with less settling time under thecontrol of the law of stable movement
Figures 8(e) and 8(f) demonstrate that the immersiondepth ℎ
1015840 of the fin and the corresponding planing force 119865119901
are both 0 This indicates that the fin is inside the cavity anddoes not have any contact with the cavity the vehicle is in astable navigation state
When 120590 is equal to 00315 the feedback control laws are120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= 3119911 which corresponds to the
point (00315 3) in the green periodic oscillation region inFigure 2
Figure 9(a) tells us that the phase trajectory is a limitcycle with period 2 when the control gain of the fin deflectionangle 119896 is equal to 3 The Lyapunov exponent spectrumcorresponding to 119896 = 3 is shown in Figure 9(b) It is notdifficult to find that the system approximately has a zeroLyapunov exponent and three negative Lyapunov exponents
The motion state of the supercavitating vehicle is shownin Figure 10 in which the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate velocity 119902 oscillate periodically at theequilibrium point (00416 13937 00190 0)
The immersion depth ℎ1015840 of the fin oscillates periodically
in the range of [0 006] (in m) which indicates that thevehicle continuously collides with the cavity wall
The fin is inside the cavity at times and does not come incontact with the cavity which results in zero planing force119865119901 The fin penetrates the cavity into the water at times
and produces the planing force oscillating periodically in therange of [0 48] (in N) The above actions repeat again andagain and such phenomenon is referred to as ldquofin attackphenomenonrdquo It also indicates that the vehicle is in anunstable periodic oscillating state
Shock and Vibration 9
0 05 10minus006
minus004
minus002
0
002
004
006
008
010
z(m
)
t (s)
(a) Vertical position
0 05 10t (s)
minus3
minus2
minus1
0
1
2
3
w(m
middotsminus1)
(b) Transverse speed
0 05 10minus0004
minus0002
0
0002
0004
t (s)
120579(r
ad)
(c) Pitch angle
0 05 10minus10
minus05
0
05
10
t (s)
q(r
ads
)
(d) Pitch rate
0 05 10minus002
0
002
004
006
008
t (s)
hp
(m)
(e) Immersion depth
0 05 10minus60
minus40
minus20
0
20
40
60
80
t (s)
Fp
(N)
(f) Planing force
Figure 12 The motion state of the system when 119896 = 55
When 120590 is equal to 00315 the feedback control laws of120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= minus55119911 are selected which
corresponds to the point (00315 minus55) in the yellow chaoticarea in Figure 2
Figure 11(a) demonstrates that the phase trajectory is achaotic attractor when the control gain of the fin deflectionangle 119896 is equal to minus55 which indicates that the chaoshas occurred and the movement of the vehicle has the
characteristics of a nonlinear dynamic behavior The Lya-punov exponent spectrum corresponding to 119896 = minus55 isgiven in Figure 11(b) It is relatively easy to find that thesystem has a positive Lyapunov exponent and three negativeLyapunov exponents suggesting that the system is in a four-dimensional chaotic state
The motion state of the system is presented in Figure 12After the launch of the supercavitating vehicle the four
10 Shock and Vibration
system state variables namely 119911119908 120579 and 119902 are in the intensenonperiodic oscillating stateThus the vehicle in motion willexperience instability
It can be observed fromFigures 12(e) and 12(f) that withinthe time range of [1 2] (values in 119904) the fin of the vehicleis continuously into contact with the cavity wall under theaction of gravity and produces the planing forceThe planingforce increases gradually with the increase of immersiondepth and the vehicle will rebound inside the cavity whichresults in the loss of the planing force The above actionsrepeat subsequently and the planing force oscillates Theexistence of the planing force will cause vibration and impactto the vehicle resulting in the loss of stability of the vehicleTherefore precise control must be exerted on the vehicle toavoid the above situations [16 17]
5 Conclusions
Thenonlinear dynamic characteristicmovement states underdifferent control parameters of the supercavitating vehiclewere analyzed based on a four-dimensional dynamical modelof the vehicle The following conclusions have been mainlyderived
(1) The movement trajectories of the supercavitatingvehicle have complicated dynamical behavior thesystem will experience Hopf bifurcation periodicalwindows chaos and other nonlinear phenomenawhen the control parameters vary
(2) The movement state of the vehicle under differentcontrol parameters was numerically and preciselyanalyzed according to the phase trajectory diagramthe bifurcation diagram and the Lyapunov exponen-tial spectrum
(3) Most importantly the authors were the first tofind that the range of parameters of the vehicle inany movement state can be determined by a two-dimensional bifurcation diagram The importance ofselecting appropriate control parameters to realize thestable navigation of the supercavitating vehicle wasdemonstrated
It is believed that the work presented in this paper is ofgreat importance for further studies on the stable controlof the underwater supercavitating vehicles especially forengineering practice
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 11472163 11402116 and 61473340)
References
[1] G Wang and M Ostoja-Starzewski ldquoLarge eddy simulationof a sheetcloud cavitation on a NACA0015 hydrofoilrdquo AppliedMathematical Modelling vol 31 no 3 pp 417ndash447 2007
[2] Q T Li Y S He and L P Xue ldquoA numerical simulationof pitching motion of the ventilated supercaviting vehicleapproximately its frontrdquo Chinese Journal of Hydrodynamics vol26 no 6 pp 589ndash685 2011
[3] A K Singhal M M Athavale H Li and Y Jiang ldquoMathemati-cal basis and validation of the full cavitation modelrdquo Journal ofFluids Engineering vol 124 no 3 pp 617ndash624 2002
[4] Y N Savchenko ldquoSupercavitation problems and perspectivesrdquoin Proceedings of the 4th International Symposium onCavitationPasadena Calif USA April 2001
[5] M A Hassouneh and E H Abed ldquoLyapunov and LMI analysisand feedback control of border collision bifurcationsrdquo Nonlin-ear Dynamics vol 50 no 3 pp 373ndash386 2007
[6] Y J Wei J H Wang J Z Zhang W Cao and W H HuangldquoNonlinear dynamics and control of underwater supercavitat-ing vehiclerdquo Journal of Vibration And Shock vol 28 no 6 pp179ndash204 2009
[7] S S Kulkarni and R Pratap ldquoStudies on the dynamics ofa supercavitating projectilerdquo Applied Mathematical Modellingvol 24 no 2 pp 113ndash129 2000
[8] J-Y Choi M Ruzzene and O A Bauchau ldquoDynamic anal-ysis of flexible supercavitating vehicles using modal-basedelementsrdquo Simulation vol 80 no 11 pp 619ndash633 2004
[9] B Feeny ldquoA nonsmooth Coulomb friction oscillatorrdquo Physica DNonlinear Phenomena vol 59 no 1ndash3 pp 25ndash38 1992
[10] A D Vasin and E V Paryshev ldquoImmersion of cylinder in a fluidthrough a cylindrical free surfacerdquo Fluid Dynamics vol 36 no2 pp 169ndash177 2001
[11] G Lin B Balachandran and E Abed ldquoSupercavitating bodydynamics bifurcations and controlrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 691ndash696 PortlandOre USA June 2005
[12] J Dzielski and A Kurdila ldquoA benchmark control problemfor supercavitating vehicles and an initial investigation ofsolutionsrdquo Journal of Vibration amp Control vol 9 no 7 pp 791ndash804 2003
[13] G J Lin B Balakumar and H A Eysd ldquoBifurcation behaviorof a supercavitating vehiclerdquo in Proceedings of the ASMEInternational Mechanical Engineering Congress and Expositionpp 293ndash300 Chicago Ill USA November 2006
[14] G J Lin B Balachandran and E H Abed ldquoDynamics andcontrol of supercavitating vehiclesrdquo Journal of Dynamic SystemsMeasurement and Control vol 130 no 2 Article ID 021003 pp281ndash287 2008
[15] M A Hassouneh V Nguyen B Balachandran and E HAbed ldquoStability analysis and control of supercavitating vehicleswith advection delayrdquo Journal of Computational and NonlinearDynamics vol 8 no 2 Article ID 021003 2012
[16] G Lin B Balachandran and E H Abed ldquoNonlinear dynamicsand bifurcations of a supercavitating vehiclerdquo IEEE Journal ofOceanic Engineering vol 32 no 4 pp 753ndash761 2007
[17] J H Wang Y J Wei and K P Yu ldquoModeling and control ofunderwater supercavitating vehicle based on memory effect ofcavityrdquo Journal of Vibration And Shock vol 29 no 8 pp 160ndash163 2010
Figure 10 The motion state of the system when 119896 = 5
is set as 120590 = 00315 the cavitator deflection angle is 120575119888=
15119911 minus 30120579 minus 03119902 and the fin deflection angle is 120575119890= 119896119911 The
bifurcation diagram of the supercavitating vehicle betweenthe system state variable 119908 and the control gain 119896 of the findeflection angle is presented in Figure 5 According to thedynamical behavior distribution presented in Figure 2 the
effective range of the control gain 119896 for the supercavitatingvehicle is presented in Figure 5 when 120590 = 00315
Figure 5 shows that when 119896 falls within the wider range[minus7614 minus5238] the system is in a chaotic state and finallythe period-2 trajectory occurs through period doublingbifurcation
8 Shock and Vibration
minus04 minus02 0 02 04 06minus20
minus10
0
10
20
q (rads)
w(m
middotsminus1)
(a)
0 5 10 15
minus100
minus80
minus60
minus40
minus20
0
20
Lyap
unov
expo
nent
spec
trum
t (s)
(b)
Figure 11 (a) Phase trajectory diagram in 119908 minus 120579 plane when 119896 = minus55 and (b) Lyapunov exponent spectrum
When 119896 is approximately 856 the periodic state endsand the system is in the divergent state The correspondingmagnified part in Figure 5 for minus80 lt 119896 lt minus50 is givenin Figure 6 which indicates that the system changes from aperiodic state into a chaotic state when 119896 = minus779
When 119896 is approximately minus7754 the tangent bifurcationoccurs leading to an intermittent chaos and forming theperiod-3 windows and then three stable periodic trajectories
The period doubling bifurcations occur for the threetrajectories when 119896 is approximately minus77 minus7616 or minus755When 119896 is approximately minus746 the secondary chaotic bandand the unstable periodic trajectories converge into chaos
When 119896 is approximately minus7358 the tangent bifurcationoccurs The system suddenly switches from chaotic state toperiodic state and the period-three windows forms With theoccurrence of a series of period doubling bifurcation thesystem enters into the chaotic state again when 119896 is betweenminus7118 andminus5264 uponwhich the system switches back fromthe chaotic state to the periodic state
4 Movement Characteristic Analysis of theUnderwater Supercavitating Vehicle
When the system of the supercavitating vehicle is not con-trolled the movement state of the system is unstable [12 14]To investigate its movement characteristics alone accordingto the two-dimensional bifurcation diagram the rest of theparameters of the system should be kept constant Assumingthat 120590 is equal to 00315 the feedback control laws 120575
119888=
15119911 minus 30120579 minus 03119902 and 120575119890= minus2195119911 which corresponds to
the point (00315 minus2195) in the red stable movement area inFigure 2Thephase trajectory diagram is shown in Figure 7(a)when the control gain of the fin deflection angle 119896 is equal tominus2195
It can be observed that when 119896 = minus2195 the phasetrajectory of the system gradually stabilizes at an equilibriumpointThe Lyapunov exponent spectrum as a function of time
is presented in Figure 7(b) in which the values of the largestLyapunov exponent curve are negative within a finite time
The motion state of the supercavitating vehicle is pre-sented in Figure 8 inwhich the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate 119902 are attracted to the equilibrium point(00047 00866 00012 0) with less settling time under thecontrol of the law of stable movement
Figures 8(e) and 8(f) demonstrate that the immersiondepth ℎ
1015840 of the fin and the corresponding planing force 119865119901
are both 0 This indicates that the fin is inside the cavity anddoes not have any contact with the cavity the vehicle is in astable navigation state
When 120590 is equal to 00315 the feedback control laws are120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= 3119911 which corresponds to the
point (00315 3) in the green periodic oscillation region inFigure 2
Figure 9(a) tells us that the phase trajectory is a limitcycle with period 2 when the control gain of the fin deflectionangle 119896 is equal to 3 The Lyapunov exponent spectrumcorresponding to 119896 = 3 is shown in Figure 9(b) It is notdifficult to find that the system approximately has a zeroLyapunov exponent and three negative Lyapunov exponents
The motion state of the supercavitating vehicle is shownin Figure 10 in which the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate velocity 119902 oscillate periodically at theequilibrium point (00416 13937 00190 0)
The immersion depth ℎ1015840 of the fin oscillates periodically
in the range of [0 006] (in m) which indicates that thevehicle continuously collides with the cavity wall
The fin is inside the cavity at times and does not come incontact with the cavity which results in zero planing force119865119901 The fin penetrates the cavity into the water at times
and produces the planing force oscillating periodically in therange of [0 48] (in N) The above actions repeat again andagain and such phenomenon is referred to as ldquofin attackphenomenonrdquo It also indicates that the vehicle is in anunstable periodic oscillating state
Shock and Vibration 9
0 05 10minus006
minus004
minus002
0
002
004
006
008
010
z(m
)
t (s)
(a) Vertical position
0 05 10t (s)
minus3
minus2
minus1
0
1
2
3
w(m
middotsminus1)
(b) Transverse speed
0 05 10minus0004
minus0002
0
0002
0004
t (s)
120579(r
ad)
(c) Pitch angle
0 05 10minus10
minus05
0
05
10
t (s)
q(r
ads
)
(d) Pitch rate
0 05 10minus002
0
002
004
006
008
t (s)
hp
(m)
(e) Immersion depth
0 05 10minus60
minus40
minus20
0
20
40
60
80
t (s)
Fp
(N)
(f) Planing force
Figure 12 The motion state of the system when 119896 = 55
When 120590 is equal to 00315 the feedback control laws of120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= minus55119911 are selected which
corresponds to the point (00315 minus55) in the yellow chaoticarea in Figure 2
Figure 11(a) demonstrates that the phase trajectory is achaotic attractor when the control gain of the fin deflectionangle 119896 is equal to minus55 which indicates that the chaoshas occurred and the movement of the vehicle has the
characteristics of a nonlinear dynamic behavior The Lya-punov exponent spectrum corresponding to 119896 = minus55 isgiven in Figure 11(b) It is relatively easy to find that thesystem has a positive Lyapunov exponent and three negativeLyapunov exponents suggesting that the system is in a four-dimensional chaotic state
The motion state of the system is presented in Figure 12After the launch of the supercavitating vehicle the four
10 Shock and Vibration
system state variables namely 119911119908 120579 and 119902 are in the intensenonperiodic oscillating stateThus the vehicle in motion willexperience instability
It can be observed fromFigures 12(e) and 12(f) that withinthe time range of [1 2] (values in 119904) the fin of the vehicleis continuously into contact with the cavity wall under theaction of gravity and produces the planing forceThe planingforce increases gradually with the increase of immersiondepth and the vehicle will rebound inside the cavity whichresults in the loss of the planing force The above actionsrepeat subsequently and the planing force oscillates Theexistence of the planing force will cause vibration and impactto the vehicle resulting in the loss of stability of the vehicleTherefore precise control must be exerted on the vehicle toavoid the above situations [16 17]
5 Conclusions
Thenonlinear dynamic characteristicmovement states underdifferent control parameters of the supercavitating vehiclewere analyzed based on a four-dimensional dynamical modelof the vehicle The following conclusions have been mainlyderived
(1) The movement trajectories of the supercavitatingvehicle have complicated dynamical behavior thesystem will experience Hopf bifurcation periodicalwindows chaos and other nonlinear phenomenawhen the control parameters vary
(2) The movement state of the vehicle under differentcontrol parameters was numerically and preciselyanalyzed according to the phase trajectory diagramthe bifurcation diagram and the Lyapunov exponen-tial spectrum
(3) Most importantly the authors were the first tofind that the range of parameters of the vehicle inany movement state can be determined by a two-dimensional bifurcation diagram The importance ofselecting appropriate control parameters to realize thestable navigation of the supercavitating vehicle wasdemonstrated
It is believed that the work presented in this paper is ofgreat importance for further studies on the stable controlof the underwater supercavitating vehicles especially forengineering practice
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 11472163 11402116 and 61473340)
References
[1] G Wang and M Ostoja-Starzewski ldquoLarge eddy simulationof a sheetcloud cavitation on a NACA0015 hydrofoilrdquo AppliedMathematical Modelling vol 31 no 3 pp 417ndash447 2007
[2] Q T Li Y S He and L P Xue ldquoA numerical simulationof pitching motion of the ventilated supercaviting vehicleapproximately its frontrdquo Chinese Journal of Hydrodynamics vol26 no 6 pp 589ndash685 2011
[3] A K Singhal M M Athavale H Li and Y Jiang ldquoMathemati-cal basis and validation of the full cavitation modelrdquo Journal ofFluids Engineering vol 124 no 3 pp 617ndash624 2002
[4] Y N Savchenko ldquoSupercavitation problems and perspectivesrdquoin Proceedings of the 4th International Symposium onCavitationPasadena Calif USA April 2001
[5] M A Hassouneh and E H Abed ldquoLyapunov and LMI analysisand feedback control of border collision bifurcationsrdquo Nonlin-ear Dynamics vol 50 no 3 pp 373ndash386 2007
[6] Y J Wei J H Wang J Z Zhang W Cao and W H HuangldquoNonlinear dynamics and control of underwater supercavitat-ing vehiclerdquo Journal of Vibration And Shock vol 28 no 6 pp179ndash204 2009
[7] S S Kulkarni and R Pratap ldquoStudies on the dynamics ofa supercavitating projectilerdquo Applied Mathematical Modellingvol 24 no 2 pp 113ndash129 2000
[8] J-Y Choi M Ruzzene and O A Bauchau ldquoDynamic anal-ysis of flexible supercavitating vehicles using modal-basedelementsrdquo Simulation vol 80 no 11 pp 619ndash633 2004
[9] B Feeny ldquoA nonsmooth Coulomb friction oscillatorrdquo Physica DNonlinear Phenomena vol 59 no 1ndash3 pp 25ndash38 1992
[10] A D Vasin and E V Paryshev ldquoImmersion of cylinder in a fluidthrough a cylindrical free surfacerdquo Fluid Dynamics vol 36 no2 pp 169ndash177 2001
[11] G Lin B Balachandran and E Abed ldquoSupercavitating bodydynamics bifurcations and controlrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 691ndash696 PortlandOre USA June 2005
[12] J Dzielski and A Kurdila ldquoA benchmark control problemfor supercavitating vehicles and an initial investigation ofsolutionsrdquo Journal of Vibration amp Control vol 9 no 7 pp 791ndash804 2003
[13] G J Lin B Balakumar and H A Eysd ldquoBifurcation behaviorof a supercavitating vehiclerdquo in Proceedings of the ASMEInternational Mechanical Engineering Congress and Expositionpp 293ndash300 Chicago Ill USA November 2006
[14] G J Lin B Balachandran and E H Abed ldquoDynamics andcontrol of supercavitating vehiclesrdquo Journal of Dynamic SystemsMeasurement and Control vol 130 no 2 Article ID 021003 pp281ndash287 2008
[15] M A Hassouneh V Nguyen B Balachandran and E HAbed ldquoStability analysis and control of supercavitating vehicleswith advection delayrdquo Journal of Computational and NonlinearDynamics vol 8 no 2 Article ID 021003 2012
[16] G Lin B Balachandran and E H Abed ldquoNonlinear dynamicsand bifurcations of a supercavitating vehiclerdquo IEEE Journal ofOceanic Engineering vol 32 no 4 pp 753ndash761 2007
[17] J H Wang Y J Wei and K P Yu ldquoModeling and control ofunderwater supercavitating vehicle based on memory effect ofcavityrdquo Journal of Vibration And Shock vol 29 no 8 pp 160ndash163 2010
Figure 11 (a) Phase trajectory diagram in 119908 minus 120579 plane when 119896 = minus55 and (b) Lyapunov exponent spectrum
When 119896 is approximately 856 the periodic state endsand the system is in the divergent state The correspondingmagnified part in Figure 5 for minus80 lt 119896 lt minus50 is givenin Figure 6 which indicates that the system changes from aperiodic state into a chaotic state when 119896 = minus779
When 119896 is approximately minus7754 the tangent bifurcationoccurs leading to an intermittent chaos and forming theperiod-3 windows and then three stable periodic trajectories
The period doubling bifurcations occur for the threetrajectories when 119896 is approximately minus77 minus7616 or minus755When 119896 is approximately minus746 the secondary chaotic bandand the unstable periodic trajectories converge into chaos
When 119896 is approximately minus7358 the tangent bifurcationoccurs The system suddenly switches from chaotic state toperiodic state and the period-three windows forms With theoccurrence of a series of period doubling bifurcation thesystem enters into the chaotic state again when 119896 is betweenminus7118 andminus5264 uponwhich the system switches back fromthe chaotic state to the periodic state
4 Movement Characteristic Analysis of theUnderwater Supercavitating Vehicle
When the system of the supercavitating vehicle is not con-trolled the movement state of the system is unstable [12 14]To investigate its movement characteristics alone accordingto the two-dimensional bifurcation diagram the rest of theparameters of the system should be kept constant Assumingthat 120590 is equal to 00315 the feedback control laws 120575
119888=
15119911 minus 30120579 minus 03119902 and 120575119890= minus2195119911 which corresponds to
the point (00315 minus2195) in the red stable movement area inFigure 2Thephase trajectory diagram is shown in Figure 7(a)when the control gain of the fin deflection angle 119896 is equal tominus2195
It can be observed that when 119896 = minus2195 the phasetrajectory of the system gradually stabilizes at an equilibriumpointThe Lyapunov exponent spectrum as a function of time
is presented in Figure 7(b) in which the values of the largestLyapunov exponent curve are negative within a finite time
The motion state of the supercavitating vehicle is pre-sented in Figure 8 inwhich the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate 119902 are attracted to the equilibrium point(00047 00866 00012 0) with less settling time under thecontrol of the law of stable movement
Figures 8(e) and 8(f) demonstrate that the immersiondepth ℎ
1015840 of the fin and the corresponding planing force 119865119901
are both 0 This indicates that the fin is inside the cavity anddoes not have any contact with the cavity the vehicle is in astable navigation state
When 120590 is equal to 00315 the feedback control laws are120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= 3119911 which corresponds to the
point (00315 3) in the green periodic oscillation region inFigure 2
Figure 9(a) tells us that the phase trajectory is a limitcycle with period 2 when the control gain of the fin deflectionangle 119896 is equal to 3 The Lyapunov exponent spectrumcorresponding to 119896 = 3 is shown in Figure 9(b) It is notdifficult to find that the system approximately has a zeroLyapunov exponent and three negative Lyapunov exponents
The motion state of the supercavitating vehicle is shownin Figure 10 in which the system state variables namelythe vertical position 119911 the transverse speed119908 the pitch angle120579 and the pitch rate velocity 119902 oscillate periodically at theequilibrium point (00416 13937 00190 0)
The immersion depth ℎ1015840 of the fin oscillates periodically
in the range of [0 006] (in m) which indicates that thevehicle continuously collides with the cavity wall
The fin is inside the cavity at times and does not come incontact with the cavity which results in zero planing force119865119901 The fin penetrates the cavity into the water at times
and produces the planing force oscillating periodically in therange of [0 48] (in N) The above actions repeat again andagain and such phenomenon is referred to as ldquofin attackphenomenonrdquo It also indicates that the vehicle is in anunstable periodic oscillating state
Shock and Vibration 9
0 05 10minus006
minus004
minus002
0
002
004
006
008
010
z(m
)
t (s)
(a) Vertical position
0 05 10t (s)
minus3
minus2
minus1
0
1
2
3
w(m
middotsminus1)
(b) Transverse speed
0 05 10minus0004
minus0002
0
0002
0004
t (s)
120579(r
ad)
(c) Pitch angle
0 05 10minus10
minus05
0
05
10
t (s)
q(r
ads
)
(d) Pitch rate
0 05 10minus002
0
002
004
006
008
t (s)
hp
(m)
(e) Immersion depth
0 05 10minus60
minus40
minus20
0
20
40
60
80
t (s)
Fp
(N)
(f) Planing force
Figure 12 The motion state of the system when 119896 = 55
When 120590 is equal to 00315 the feedback control laws of120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= minus55119911 are selected which
corresponds to the point (00315 minus55) in the yellow chaoticarea in Figure 2
Figure 11(a) demonstrates that the phase trajectory is achaotic attractor when the control gain of the fin deflectionangle 119896 is equal to minus55 which indicates that the chaoshas occurred and the movement of the vehicle has the
characteristics of a nonlinear dynamic behavior The Lya-punov exponent spectrum corresponding to 119896 = minus55 isgiven in Figure 11(b) It is relatively easy to find that thesystem has a positive Lyapunov exponent and three negativeLyapunov exponents suggesting that the system is in a four-dimensional chaotic state
The motion state of the system is presented in Figure 12After the launch of the supercavitating vehicle the four
10 Shock and Vibration
system state variables namely 119911119908 120579 and 119902 are in the intensenonperiodic oscillating stateThus the vehicle in motion willexperience instability
It can be observed fromFigures 12(e) and 12(f) that withinthe time range of [1 2] (values in 119904) the fin of the vehicleis continuously into contact with the cavity wall under theaction of gravity and produces the planing forceThe planingforce increases gradually with the increase of immersiondepth and the vehicle will rebound inside the cavity whichresults in the loss of the planing force The above actionsrepeat subsequently and the planing force oscillates Theexistence of the planing force will cause vibration and impactto the vehicle resulting in the loss of stability of the vehicleTherefore precise control must be exerted on the vehicle toavoid the above situations [16 17]
5 Conclusions
Thenonlinear dynamic characteristicmovement states underdifferent control parameters of the supercavitating vehiclewere analyzed based on a four-dimensional dynamical modelof the vehicle The following conclusions have been mainlyderived
(1) The movement trajectories of the supercavitatingvehicle have complicated dynamical behavior thesystem will experience Hopf bifurcation periodicalwindows chaos and other nonlinear phenomenawhen the control parameters vary
(2) The movement state of the vehicle under differentcontrol parameters was numerically and preciselyanalyzed according to the phase trajectory diagramthe bifurcation diagram and the Lyapunov exponen-tial spectrum
(3) Most importantly the authors were the first tofind that the range of parameters of the vehicle inany movement state can be determined by a two-dimensional bifurcation diagram The importance ofselecting appropriate control parameters to realize thestable navigation of the supercavitating vehicle wasdemonstrated
It is believed that the work presented in this paper is ofgreat importance for further studies on the stable controlof the underwater supercavitating vehicles especially forengineering practice
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 11472163 11402116 and 61473340)
References
[1] G Wang and M Ostoja-Starzewski ldquoLarge eddy simulationof a sheetcloud cavitation on a NACA0015 hydrofoilrdquo AppliedMathematical Modelling vol 31 no 3 pp 417ndash447 2007
[2] Q T Li Y S He and L P Xue ldquoA numerical simulationof pitching motion of the ventilated supercaviting vehicleapproximately its frontrdquo Chinese Journal of Hydrodynamics vol26 no 6 pp 589ndash685 2011
[3] A K Singhal M M Athavale H Li and Y Jiang ldquoMathemati-cal basis and validation of the full cavitation modelrdquo Journal ofFluids Engineering vol 124 no 3 pp 617ndash624 2002
[4] Y N Savchenko ldquoSupercavitation problems and perspectivesrdquoin Proceedings of the 4th International Symposium onCavitationPasadena Calif USA April 2001
[5] M A Hassouneh and E H Abed ldquoLyapunov and LMI analysisand feedback control of border collision bifurcationsrdquo Nonlin-ear Dynamics vol 50 no 3 pp 373ndash386 2007
[6] Y J Wei J H Wang J Z Zhang W Cao and W H HuangldquoNonlinear dynamics and control of underwater supercavitat-ing vehiclerdquo Journal of Vibration And Shock vol 28 no 6 pp179ndash204 2009
[7] S S Kulkarni and R Pratap ldquoStudies on the dynamics ofa supercavitating projectilerdquo Applied Mathematical Modellingvol 24 no 2 pp 113ndash129 2000
[8] J-Y Choi M Ruzzene and O A Bauchau ldquoDynamic anal-ysis of flexible supercavitating vehicles using modal-basedelementsrdquo Simulation vol 80 no 11 pp 619ndash633 2004
[9] B Feeny ldquoA nonsmooth Coulomb friction oscillatorrdquo Physica DNonlinear Phenomena vol 59 no 1ndash3 pp 25ndash38 1992
[10] A D Vasin and E V Paryshev ldquoImmersion of cylinder in a fluidthrough a cylindrical free surfacerdquo Fluid Dynamics vol 36 no2 pp 169ndash177 2001
[11] G Lin B Balachandran and E Abed ldquoSupercavitating bodydynamics bifurcations and controlrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 691ndash696 PortlandOre USA June 2005
[12] J Dzielski and A Kurdila ldquoA benchmark control problemfor supercavitating vehicles and an initial investigation ofsolutionsrdquo Journal of Vibration amp Control vol 9 no 7 pp 791ndash804 2003
[13] G J Lin B Balakumar and H A Eysd ldquoBifurcation behaviorof a supercavitating vehiclerdquo in Proceedings of the ASMEInternational Mechanical Engineering Congress and Expositionpp 293ndash300 Chicago Ill USA November 2006
[14] G J Lin B Balachandran and E H Abed ldquoDynamics andcontrol of supercavitating vehiclesrdquo Journal of Dynamic SystemsMeasurement and Control vol 130 no 2 Article ID 021003 pp281ndash287 2008
[15] M A Hassouneh V Nguyen B Balachandran and E HAbed ldquoStability analysis and control of supercavitating vehicleswith advection delayrdquo Journal of Computational and NonlinearDynamics vol 8 no 2 Article ID 021003 2012
[16] G Lin B Balachandran and E H Abed ldquoNonlinear dynamicsand bifurcations of a supercavitating vehiclerdquo IEEE Journal ofOceanic Engineering vol 32 no 4 pp 753ndash761 2007
[17] J H Wang Y J Wei and K P Yu ldquoModeling and control ofunderwater supercavitating vehicle based on memory effect ofcavityrdquo Journal of Vibration And Shock vol 29 no 8 pp 160ndash163 2010
Figure 12 The motion state of the system when 119896 = 55
When 120590 is equal to 00315 the feedback control laws of120575119888= 15119911 minus 30120579 minus 03119902 and 120575
119890= minus55119911 are selected which
corresponds to the point (00315 minus55) in the yellow chaoticarea in Figure 2
Figure 11(a) demonstrates that the phase trajectory is achaotic attractor when the control gain of the fin deflectionangle 119896 is equal to minus55 which indicates that the chaoshas occurred and the movement of the vehicle has the
characteristics of a nonlinear dynamic behavior The Lya-punov exponent spectrum corresponding to 119896 = minus55 isgiven in Figure 11(b) It is relatively easy to find that thesystem has a positive Lyapunov exponent and three negativeLyapunov exponents suggesting that the system is in a four-dimensional chaotic state
The motion state of the system is presented in Figure 12After the launch of the supercavitating vehicle the four
10 Shock and Vibration
system state variables namely 119911119908 120579 and 119902 are in the intensenonperiodic oscillating stateThus the vehicle in motion willexperience instability
It can be observed fromFigures 12(e) and 12(f) that withinthe time range of [1 2] (values in 119904) the fin of the vehicleis continuously into contact with the cavity wall under theaction of gravity and produces the planing forceThe planingforce increases gradually with the increase of immersiondepth and the vehicle will rebound inside the cavity whichresults in the loss of the planing force The above actionsrepeat subsequently and the planing force oscillates Theexistence of the planing force will cause vibration and impactto the vehicle resulting in the loss of stability of the vehicleTherefore precise control must be exerted on the vehicle toavoid the above situations [16 17]
5 Conclusions
Thenonlinear dynamic characteristicmovement states underdifferent control parameters of the supercavitating vehiclewere analyzed based on a four-dimensional dynamical modelof the vehicle The following conclusions have been mainlyderived
(1) The movement trajectories of the supercavitatingvehicle have complicated dynamical behavior thesystem will experience Hopf bifurcation periodicalwindows chaos and other nonlinear phenomenawhen the control parameters vary
(2) The movement state of the vehicle under differentcontrol parameters was numerically and preciselyanalyzed according to the phase trajectory diagramthe bifurcation diagram and the Lyapunov exponen-tial spectrum
(3) Most importantly the authors were the first tofind that the range of parameters of the vehicle inany movement state can be determined by a two-dimensional bifurcation diagram The importance ofselecting appropriate control parameters to realize thestable navigation of the supercavitating vehicle wasdemonstrated
It is believed that the work presented in this paper is ofgreat importance for further studies on the stable controlof the underwater supercavitating vehicles especially forengineering practice
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 11472163 11402116 and 61473340)
References
[1] G Wang and M Ostoja-Starzewski ldquoLarge eddy simulationof a sheetcloud cavitation on a NACA0015 hydrofoilrdquo AppliedMathematical Modelling vol 31 no 3 pp 417ndash447 2007
[2] Q T Li Y S He and L P Xue ldquoA numerical simulationof pitching motion of the ventilated supercaviting vehicleapproximately its frontrdquo Chinese Journal of Hydrodynamics vol26 no 6 pp 589ndash685 2011
[3] A K Singhal M M Athavale H Li and Y Jiang ldquoMathemati-cal basis and validation of the full cavitation modelrdquo Journal ofFluids Engineering vol 124 no 3 pp 617ndash624 2002
[4] Y N Savchenko ldquoSupercavitation problems and perspectivesrdquoin Proceedings of the 4th International Symposium onCavitationPasadena Calif USA April 2001
[5] M A Hassouneh and E H Abed ldquoLyapunov and LMI analysisand feedback control of border collision bifurcationsrdquo Nonlin-ear Dynamics vol 50 no 3 pp 373ndash386 2007
[6] Y J Wei J H Wang J Z Zhang W Cao and W H HuangldquoNonlinear dynamics and control of underwater supercavitat-ing vehiclerdquo Journal of Vibration And Shock vol 28 no 6 pp179ndash204 2009
[7] S S Kulkarni and R Pratap ldquoStudies on the dynamics ofa supercavitating projectilerdquo Applied Mathematical Modellingvol 24 no 2 pp 113ndash129 2000
[8] J-Y Choi M Ruzzene and O A Bauchau ldquoDynamic anal-ysis of flexible supercavitating vehicles using modal-basedelementsrdquo Simulation vol 80 no 11 pp 619ndash633 2004
[9] B Feeny ldquoA nonsmooth Coulomb friction oscillatorrdquo Physica DNonlinear Phenomena vol 59 no 1ndash3 pp 25ndash38 1992
[10] A D Vasin and E V Paryshev ldquoImmersion of cylinder in a fluidthrough a cylindrical free surfacerdquo Fluid Dynamics vol 36 no2 pp 169ndash177 2001
[11] G Lin B Balachandran and E Abed ldquoSupercavitating bodydynamics bifurcations and controlrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 691ndash696 PortlandOre USA June 2005
[12] J Dzielski and A Kurdila ldquoA benchmark control problemfor supercavitating vehicles and an initial investigation ofsolutionsrdquo Journal of Vibration amp Control vol 9 no 7 pp 791ndash804 2003
[13] G J Lin B Balakumar and H A Eysd ldquoBifurcation behaviorof a supercavitating vehiclerdquo in Proceedings of the ASMEInternational Mechanical Engineering Congress and Expositionpp 293ndash300 Chicago Ill USA November 2006
[14] G J Lin B Balachandran and E H Abed ldquoDynamics andcontrol of supercavitating vehiclesrdquo Journal of Dynamic SystemsMeasurement and Control vol 130 no 2 Article ID 021003 pp281ndash287 2008
[15] M A Hassouneh V Nguyen B Balachandran and E HAbed ldquoStability analysis and control of supercavitating vehicleswith advection delayrdquo Journal of Computational and NonlinearDynamics vol 8 no 2 Article ID 021003 2012
[16] G Lin B Balachandran and E H Abed ldquoNonlinear dynamicsand bifurcations of a supercavitating vehiclerdquo IEEE Journal ofOceanic Engineering vol 32 no 4 pp 753ndash761 2007
[17] J H Wang Y J Wei and K P Yu ldquoModeling and control ofunderwater supercavitating vehicle based on memory effect ofcavityrdquo Journal of Vibration And Shock vol 29 no 8 pp 160ndash163 2010
system state variables namely 119911119908 120579 and 119902 are in the intensenonperiodic oscillating stateThus the vehicle in motion willexperience instability
It can be observed fromFigures 12(e) and 12(f) that withinthe time range of [1 2] (values in 119904) the fin of the vehicleis continuously into contact with the cavity wall under theaction of gravity and produces the planing forceThe planingforce increases gradually with the increase of immersiondepth and the vehicle will rebound inside the cavity whichresults in the loss of the planing force The above actionsrepeat subsequently and the planing force oscillates Theexistence of the planing force will cause vibration and impactto the vehicle resulting in the loss of stability of the vehicleTherefore precise control must be exerted on the vehicle toavoid the above situations [16 17]
5 Conclusions
Thenonlinear dynamic characteristicmovement states underdifferent control parameters of the supercavitating vehiclewere analyzed based on a four-dimensional dynamical modelof the vehicle The following conclusions have been mainlyderived
(1) The movement trajectories of the supercavitatingvehicle have complicated dynamical behavior thesystem will experience Hopf bifurcation periodicalwindows chaos and other nonlinear phenomenawhen the control parameters vary
(2) The movement state of the vehicle under differentcontrol parameters was numerically and preciselyanalyzed according to the phase trajectory diagramthe bifurcation diagram and the Lyapunov exponen-tial spectrum
(3) Most importantly the authors were the first tofind that the range of parameters of the vehicle inany movement state can be determined by a two-dimensional bifurcation diagram The importance ofselecting appropriate control parameters to realize thestable navigation of the supercavitating vehicle wasdemonstrated
It is believed that the work presented in this paper is ofgreat importance for further studies on the stable controlof the underwater supercavitating vehicles especially forengineering practice
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 11472163 11402116 and 61473340)
References
[1] G Wang and M Ostoja-Starzewski ldquoLarge eddy simulationof a sheetcloud cavitation on a NACA0015 hydrofoilrdquo AppliedMathematical Modelling vol 31 no 3 pp 417ndash447 2007
[2] Q T Li Y S He and L P Xue ldquoA numerical simulationof pitching motion of the ventilated supercaviting vehicleapproximately its frontrdquo Chinese Journal of Hydrodynamics vol26 no 6 pp 589ndash685 2011
[3] A K Singhal M M Athavale H Li and Y Jiang ldquoMathemati-cal basis and validation of the full cavitation modelrdquo Journal ofFluids Engineering vol 124 no 3 pp 617ndash624 2002
[4] Y N Savchenko ldquoSupercavitation problems and perspectivesrdquoin Proceedings of the 4th International Symposium onCavitationPasadena Calif USA April 2001
[5] M A Hassouneh and E H Abed ldquoLyapunov and LMI analysisand feedback control of border collision bifurcationsrdquo Nonlin-ear Dynamics vol 50 no 3 pp 373ndash386 2007
[6] Y J Wei J H Wang J Z Zhang W Cao and W H HuangldquoNonlinear dynamics and control of underwater supercavitat-ing vehiclerdquo Journal of Vibration And Shock vol 28 no 6 pp179ndash204 2009
[7] S S Kulkarni and R Pratap ldquoStudies on the dynamics ofa supercavitating projectilerdquo Applied Mathematical Modellingvol 24 no 2 pp 113ndash129 2000
[8] J-Y Choi M Ruzzene and O A Bauchau ldquoDynamic anal-ysis of flexible supercavitating vehicles using modal-basedelementsrdquo Simulation vol 80 no 11 pp 619ndash633 2004
[9] B Feeny ldquoA nonsmooth Coulomb friction oscillatorrdquo Physica DNonlinear Phenomena vol 59 no 1ndash3 pp 25ndash38 1992
[10] A D Vasin and E V Paryshev ldquoImmersion of cylinder in a fluidthrough a cylindrical free surfacerdquo Fluid Dynamics vol 36 no2 pp 169ndash177 2001
[11] G Lin B Balachandran and E Abed ldquoSupercavitating bodydynamics bifurcations and controlrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 691ndash696 PortlandOre USA June 2005
[12] J Dzielski and A Kurdila ldquoA benchmark control problemfor supercavitating vehicles and an initial investigation ofsolutionsrdquo Journal of Vibration amp Control vol 9 no 7 pp 791ndash804 2003
[13] G J Lin B Balakumar and H A Eysd ldquoBifurcation behaviorof a supercavitating vehiclerdquo in Proceedings of the ASMEInternational Mechanical Engineering Congress and Expositionpp 293ndash300 Chicago Ill USA November 2006
[14] G J Lin B Balachandran and E H Abed ldquoDynamics andcontrol of supercavitating vehiclesrdquo Journal of Dynamic SystemsMeasurement and Control vol 130 no 2 Article ID 021003 pp281ndash287 2008
[15] M A Hassouneh V Nguyen B Balachandran and E HAbed ldquoStability analysis and control of supercavitating vehicleswith advection delayrdquo Journal of Computational and NonlinearDynamics vol 8 no 2 Article ID 021003 2012
[16] G Lin B Balachandran and E H Abed ldquoNonlinear dynamicsand bifurcations of a supercavitating vehiclerdquo IEEE Journal ofOceanic Engineering vol 32 no 4 pp 753ndash761 2007
[17] J H Wang Y J Wei and K P Yu ldquoModeling and control ofunderwater supercavitating vehicle based on memory effect ofcavityrdquo Journal of Vibration And Shock vol 29 no 8 pp 160ndash163 2010
