Numerical Modeling and Simulation of Wave Impact of a ...2 ModellingandSimulationinEngineering Navier-Stokessolver[12].PengandWeisimulatedthewater entry of a circular cylinder based

Research ArticleNumerical Modeling and Simulation of Wave Impact ofa Circular Cylinder during the Submergence Process

Xiaozhou Hu Yiyao Jiang and Daojun Cai

College of Mechanical and Electrical Engineering Central South University Changsha 410083 China

Correspondence should be addressed to Xiaozhou Hu smallboathu163com

Received 19 April 2017 Revised 12 August 2017 Accepted 2 October 2017 Published 3 December 2017

Academic Editor Dimitrios E Manolakos

Copyright copy 2017 Xiaozhou Hu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Wave slamming loads on a circular cylinder during water entry and the subsequence submergence process are predicted basedon a numerical wave load model The wave impact problems are analyzed by solving Reynolds-Averaged Navier-Stokes (RANS)equations andVOF equations A finite volume approach (FV) is employed to implement the discretization of the RANS equations Atwo-dimensional numerical wave tank is established to simulate regular oceanwavesThewave slamming problems are investigatedby deploying a circular cylinder into waves with a constant vertical velocityThe present numerical method is validated using othernumerical or theoretical results in accordance with varying free surface profiles when a circular cylinder sinks in calm water Anumerical example is given to show the submergence process of the circular cylinder in waves and both free surface profiles andthe pressure distributions on the cylinder of different time instants are obtained Time histories of hydrodynamic load on thecylinder during the submergence process for different wave impact angles wave heights and wave periods are obtained and resultsare analyzed in detail

1 Introduction

Wave impact problem is of great interest in the marineand offshore industries especially for subsea working sys-tems which are widely used in such applications as marineresource development and utilization maritime explorationand survey the important problem which needs to be solvedbefore their proper operations is to accurately study thephenomenon of offshore structures impacting ocean waveand the following submergence processTheprocess of subseastructures lowering through wave zone accompanies theinteraction between the air wave and the solid body which isa complicated fluid-solid problem involving consideration oftime-varying hydrodynamic forces (slamming drag inertiaand buoyancy) and time-varying waves In hostile oceanconditions these forces can result in significant localized andeven catastrophic structural damage on structures rangingfrom deployed structures to deploying equipment and heavecompensator systems and so forth Therefore the accurateprediction of wave slamming loads and time histories ofhydrodynamic force as well as the sensitivity of these loadsto wave parameters is of significant importance

Researches on water entry and wave impact problemswere firstly carried out by Karman [1] who studied loads onseaplane floats shaped as wedges during water entry Wagnerdeveloped methods of Karman by taking into account thepiled-up water surface along the side of the body [2]

For a rigid cylinder pioneering studies applied severaldifferent methods including flat plate theories generalizedWagner theory [3] boundary element method (BEM) [45] and Constrained Interpolation Profile (CIP) method [67] In recent years researchers applied other computationalfluid dynamics (CFD) methods to this field For exampleZhang et al numerically simulated the water entry of acylinder based on Lattice Boltzmann Method (LBM) [8]Vandamme et al simulated water entry and exit of a cylinderwith a weakly compressible smoothed particle hydrodynam-ics (SPH) method [9] Skillen et al used SPH method toinvestigate the motion of a circular cylinder dropping ontoinitially still water numerically [10] Gu et al simulatedwater impact problem of a semicircular cylinder with thefree surface captured using level set method [11] Nguyenet al studied water entry of a circular cylinder by integrat-ing a moving Chimera grid method to a preconditioned

HindawiModelling and Simulation in EngineeringVolume 2017 Article ID 2197150 14 pageshttpsdoiorg10115520172197150

2 Modelling and Simulation in Engineering

Navier-Stokes solver [12] Peng and Wei simulated the waterentry of a circular cylinder based on a CIP method withthe parallel algorithm [13] Iranmanesh and Passandideh-Fard numerically investigated water entry of a horizontal cir-cular cylinder for low Froude numbers by the combinationof the fast-fictitious-domain method and the volume-of-fluid (VOF) technique [14] NAIR and TOMAR employedan incompressible SmoothedParticleHydrodynamics (ISPH)method to simulate water entry of 2D circular cylinders[15] Aristodemo et al performed numerical study on wave-induced forces of submerged circular cylinders by SPHmethod [16] Advances in computer technology and CFDhave made it possible to use commercial CFD codes to solvewave impact problem For example Mnasri et al used Fluentcode with a moving grid to analyze the free surface evolutioninduced by one or two moving cylinders [17] Chen et alstudied water entry of a horizontal cylinder based on VOFmethod by Fluent [18] Ghadimi et al utilized FLOW-3Dcodeto study thewater entry of a circular cylinder and conducted acomparison between the linear and nonlinear solutions [19]Tassin et al investigated water impact problem of a body withtime-varying shape by CFD code OpenFOAM [20] Larsenused the CFD code STAR-CCM+ to calculate impact loadson circular cylinders during water entry [21]

This work is devoted to investigate the complex inter-action between a circular cylinder and waves during thesubmergence process Firstly both governing equations andboundary conditions are outlined to study the interactionbetween wave and cylinders a numerical wave tank isestablished and numerical wave generation and absorptionmethod are presented Secondly snapshots of the simulationof calm water entry of a circular cylinder are compared withthose of experiments by previous researchesThen numericalsimulations of a circular cylinder impact with waves areconducted and free surface profiles of different time instantsof submergence process are shown Finally slamming loadson the circular cylinder are computed and influence of waveparameters is discussed

2 The Numerical Method

21 Governing Equations In this work governing equationsfor the CFD calculations are RANS equations for homoge-neous incompressible fluid flows and they are written asfollows

120597119906119894120597119909119894 = 0 (119894 = 1 2) 120597 (120588119906119894)120597119905 + 120597 (120588119906119894119906119895)120597119909119894 = minus 120597119901120597119909119894

+ 120597120597119909119895 [120583(120597119906119894120597119909119895 +

120597119906119895120597119909119894 )]

+ 120597120597119909119895 (minus12058811990610158401198941199061015840119895) + 120588119891119894

(1)

where 119894 119895 = 1 2 for two-dimensional flows 119906119894 is the averagevelocity of 119894th coordinate axis 120588 is the density119901 is the averagepressure120583 is the dynamic viscous coefficient119891119894 is the averagebody force minus12058811990610158401198941199061015840119895 is the Reynolds stress120597 (120588119896)120597119905 +

120597 (120588119896119906119895)120597119909119894 = 120597120597119909119895 [(120583 +

120583119905120590119896)120597119896120597119909119895] + 120588119866119896

minus 120588120576120597 (120588120576)120597119905 +

120597 (120588120576119906119895)120597119909119894 = 120597120597119909119895 [(120583 +

120583119905120590120576)120597119896120597119909119895]

+ 120576119896 (1198621205761120588119866119896 minus 1198621205762120588120576)

120583119905 = 1198621205831205881198962

120576

119866119896 = 120583119905120588 (120597119906119894120597119909119895 +

120597119906119895120597119909119894 )

120597119906119894120597119909119895

(2)

where 119896 is the turbulence kinetic energy 120576 is the dissipationrate of turbulence kinetic energy 120583119905 is the eddy viscosity 119866119896is the turbulent energy production

The empirical coefficients in (2) are given as follows [22]

119862120583 = 0091198621205761 = 1441198621205762 = 192120590119896 = 10120590120576 = 13

(3)

The complex free surface is tracked by the VOF methodand to accomplish the capturing of the interface between theair phase and the water phase a continuity equation for thefraction of volume of water is solved which has the followingform

120597119886119908120597119905 + 119906119894120597 (119886119908)120597119909119894 = 0 (4)

The sum of the volume fraction of water and air is givenby

119886119908 + 119886119886 = 1 (5)

where 119886119908 and 119886119886 are the volume fraction of water and air ineach cell respectively

22 Boundary Conditions To solve governing equations it isnecessary to specify appropriate boundary conditions at allboundaries of the domain The boundary conditions whichneed to be satisfied are as follows (1) the kinematic anddynamic free surface conditions at the free surface and (2)

Modelling and Simulation in Engineering 3

z

x

Dampingzone

Inputboundary

L

Wavegeneration

zone

Workingzone

H

Circular cylinder

d

Waves

L0 Ld

Ω(t)

Bottom Γb

Wall Γw

Free surface Γf

ΓI

Figure 1 Numerical wave tank and a moving circular cylinder

the no-slip boundary condition at the tank bottom andthe rigid body and (3) to investigate the dynamic problemduring a circular cylinder lowering through wave zone a 2Dnumerical wave tank is utilized The left wall boundary is awave-maker while the right part of the domain is a dampingzone as shown in Figure 1

23 Wave Generation Method and Numerical Wave Dissipa-tion Beach In order to investigate wave loads on a circularcylinder during its crossing air-water interface it is necessaryto establish a numerical wave tank In this paper a piston typewave-maker is utilized to simulate ocean waves and a sketchof a numerical wave tank with a piston wave-maker locatedat the left boundary of the domain is illustrated in Figure 1

The plunger moves horizontally with a sinusoidal func-tion

119883(119905) = 11988302 sin120596119905 (6)

where 1198830 is the maximum displacement of plunger and 120596 isthe angular frequency Then the free surface displacement isgiven by

120578 (119909 119905) = 11988302 (4 sinh2119896119889

2119896119889 + sinh 2119896119889 cos (119896119909 minus 120596119905)

+ infinsum119899=1

4 sin2 (119896119899119889)2119896119899119889 + sin (2119896119899119889)119890minus119896119899119909 sin120596119905)

(7)

where 119889 is calm water depth and 119896 is wave numberEquation (7) has two parts the first is the incoming wave

and the second part is attenuating standing wave With thesecond part of (7) eliminated surface elevation can be givenas follows

120578 (119909 119905) = 119883024 sinh2119896119889

2119896119889 + sinh 2119896119889 cos (119896119909 minus 120596119905)= 1198672 cos (119896119909 minus 120596119905)

(8)

And wave height119867 can be obtained as follows

119867 = 4 sinh2 (119896119889)2119896119889 + sinh (2119896119889)1198830 (9)

As mentioned in Section 22 to eliminate attenuatingstanding waves porous media are employed to form anartificial dissipation zone which is placed at the end of thedomain To model porous media a momentum source termis added to standard fluid flow equations which can be givenby

119878119894 = minus(120583120572V119894 + 119862212120588 1003816100381610038161003816V1198941003816100381610038161003816 V119894) (10)

where 119878119894 is the source term of momentum equations in 119894th(119909 119910 or 119911) direction |V119894| is the velocity magnitude 1198622 is theinertial resistance factor and 120572 is the permeability

24 The Grid of Numerical Model The 2D numerical tankis a rectangle with 40m height and the length of which isdetermined by the target waves The calm water depth (119889) is30m and the distance 1198710 between the center of the circularcylinder and the left boundary is 75m The numerical gridgenerated by Gambit is shown in Figure 2

A finer mesh can be observed near the free surface forthe possibility of wave breaking and near the moving wave-maker while coarser meshes can be found towards the rightwall and the bottom of domain

3 Dynamic Results and Analysis

A commercial CFD software ANSYS Fluent was employedto solve the RANS equation with the free surface capturingVOF scheme and turbulence equations Setups of modelsin the present simulations are briefly introduced as followsgeoreconstruct scheme which is the default scheme of theVOF model is applied to compute the air-water interface(wave surface) For pressure-velocity coupling the pressureimplicit with splitting of operators (PISO) scheme is usedbecause problems presented in this paper generally involvetransient flows PRESTO scheme is used for pressure inter-polation scheme of VOF two-phase models The second-order upwind scheme is applied for discretization of the


(a) General diagram of mesh

(b) Mesh surrounding moving body (c) Mesh near the wave-maker

Figure 2 Schematic diagrams of mesh

momentum equation to obtain second-order accuracy Asfar as the time step size (Δ119905) is concerned for wave entryproblem with constant velocity which is a typical transientphenomenon Δ119905 can be calculated by

Δ119905 le 119862Δ119909V119891 (11)

where 119862 is the Courant number which is set to 025 forthe VOF model Δ119909 is the minimum cell size and V119891 isthe maximum characteristic velocity in the fluid domainFor other detailed information one can refer to manuals ofANSYS Fluent [23 24]

31 Numerical Wave Tank A numerical experiment forsimulating a target wave (wave height = 25m wave period =6 s) is conducted to verify the versatility and correctness of thenumerical wave generation and damping methods presentedin Section 2 The permeability of porous media in dampingzone can be given by

1120572 (119909) = 1205720 (

119909 minus 119909119904119909119890 minus 119909119904) (12)

where 1205720 is the constant permeability which can be chosen as1times106m2 119909119904 and 119909119890 are starting point coordinate and endingpoint coordinate of damping zone respectively in this paper119909119890 = 300m and 119909119904 = 150m

Four wave elevation probes are placed at locations of 119909 =50m 100m 200m and 300m of the numerical wave tank

and the ratio of wave elevations of simulated waves to thewave amplitude of target wave (120578119860) is employed to evaluatethe capability of wave generating and wave absorbing Asshown in Figure 3 for wave elevations in the working zone(locations of 119909 = 50m and 100m) of NWT the amplitude ofthe ratio 120578119860 is approximately equal to 100 which meansthat the simulated wave is almost identical with the targetwave In the damping zone (locations of 119909 = 200m and300m) incident waves gradually vanish and at the end ofthe damping zone 120578119860 are less than 5 which indicates thatwave energy is sufficiently dissipated inside the damping zonedue to adequate damping capability Therefore a conclusioncan be drawn that the performance of the NWT can satisfythe need of studying the problem of a circular cylinderlowering through wave zone numerically It is worth notingthat for the first periods of wave elevations of locations ofworking zone the simulated waves are unstable which canbe clearly observed in Figures 3(a) and 3(b) therefore whenthe simulated waves are used to study wave-body interactionproblem the first periods need to be avoided

32 Sinking of a Circular Cylinder in Calm Water To reflectthe capability of the presented numerical model to deal withthe problem of a body moving near the air-water interfacethe example of a circular cylinder sinking in calm wateris used The setup of the numerical model is given belowThe radius of the circular cylinder 119877 is 1m and the initialdistance ℎ between its center and calmwater surface is 125mThe gravitational acceleration 119892 is 98ms2 and the cylindermoves downwards with a constant velocity V = 122ms


100

0

minus100

A

()

0 20 40 60 80 100

Time (s)

(a)

100

0

minus100

A

()

0 20 40 60 80 100

Time (s)

(b)

50

0

minus50

A

()

0 20 40 60 80 100

Time (s)

(c)

5

0

minus5

A

()

0 20 40 60 80 100

Time (s)

(d)

Figure 3 Wave elevation at four different locations (a) 119909 = 50m (b) 119909 = 100m (c) 119909 = 200m (d) 119909 = 300m

Table 1 Mesh sizes used for mesh size study

Mesh size Number of cells Mesh size near the circularcylinder Mesh size near the free surface

Mesh 1 26160 007854m (12058711987740) 02m (H125)Mesh 2 36388 005236m (12058711987760) 0133m (H1875)Mesh 3 51590 003927m (12058711987780) 01m (H25)Mesh 4 57460 003142m (120587119877100) 008m (H3125)

Parameters mentioned above render Froude number Fr =Vradic119892119877 = 039 relative distance 120576 = ℎ119877 = 125 and119879 = V119905ℎ which are the same as those used by Greenhowand Moyo [25] and Lin [26]

Comparisons between free surface profiles of the presentmodel and previous researches which include Linrsquos andGreenhow and Moyorsquos numerical results and Tyvand andMilohrsquos theoretical results [27] are shown in Figure 4 It isclearly demonstrated that there is a good agreement betweenresults of the present model and previous results whichverifies the effectiveness and accuracy of the numericalmodelpresented in this work

33 Discussion on Mesh Size The choice of mesh can affectthe accuracy of simulation results computational efficiencyand the solution stability and is dependent on amount ofmemory available Therefore it is important to discuss thesize and quality ofmesh and choose appropriatemeshes Fourdifferent mesh sizes are employed to perform the mesh sizestudy and main differences between them are sizes of finermeshes near the circular cylinder the wave-maker and thefree surface while coarser meshes are basically the sameMore information of meshes can be found in Table 1

Main setups of simulations are as follows the target wavewhich is simulated in Section 31 is employed the radius ofthe circular cylinder119877 is 1m the constant downward velocityof the cylinder V is 1ms the initial distance between thecenter of the cylinder and the calm water surface ℎ is 4mBesides the initial time of simulation 1199050 is set as 0 s and themotion of the wave-maker will generate the target wave since

the initial timeThe initial time 1198790 when the circular cylinderbegins to move downwards is set as 27684 s (119879 = V119905ℎ = 0)which means the circular cylinder would impact with thewave surface at 119879impact = 30684 s (119879 = V119905ℎ = 075)

Figure 5 shows time histories of hydrodynamic forceson the circular cylinder As can be observed from Figure 5simulation results obtained from the four different meshsizes are similar The largest difference can be found at theimpact phase where a finer mesh gives a higher impact forceFurthermore results of Mesh 3 and Mesh 4 are very close tothe value calculated from the equation given by Campbelland Weynberg [28] and the difference between results ofMesh 3 and Mesh 4 is small Therefore to obtain a highercomputational efficiency and keep accuracy of simulationresults Mesh 3 is used in the following analysis

34 An Example of the Water Entry of a Circular Cylinderin Waves An example is utilized to show the whole waterentry process of a circular cylinder in waves main setups ofthis numerical example are the same as those described inSection 33

The free surface profiles and pressure distributions at 119879 =075 0775 084 09 11 and 16 are shown in Figures 6 and7 respectively Figures can demonstrate the water entry andthe submergence process of the circular cylinder in waves

At 119879 = 075 the circular cylinder has already impactswith the wave surface which indicates that the impact timeof simulation is a little earlier than the theoretical impacttime due to simulation errors From Figures 6(a) and 7(a)the following phenomenon can be observed firstly pressure


1

0

minus1

minus2

minus3minus2 minus1 0 1 2

xd

yd

(a) 119879 = 00

1

0

minus1

minus2


xd

yd

(b) 119879 = 04

1

0

minus1

minus2


xd

yd

(c) 119879 = 10

1

0

minus1

minus2


xd

yd

(d) 119879 = 20

Figure 4 Comparisons of results by different models of a circular cylinder sinking in calm water the present results (red solid line) Linrsquosresults (blue solid line) Greenhow et alrsquos results (dashed line) Tyvand et alrsquos results (dotted line)

distributions on the cylinder are nonsymmetric due toinfluence of waves secondly the peak pressure appears attwo contact points between the body and the water-air two-phase fluid and the right peak pressure is bigger than theleft one because regular waves travel from the left boundarybesides the pressure on the bottom point is smaller than thatof contact points thirdly the pressure on parts of the cylindergetting in contact with water is much bigger than that with airwhich is approximately equal to zero The partly immersedstate of the circular cylinder at 119879 = 0775 is shown in Figures6(b)ndash6(d) and 7(b)ndash7(d) a noteworthy feature is the negativepressure that appears at the dry part of the cylinder which isrelated to hydroelasticity and needs further research Figures6(e) and 7(e) reflect that the cylinder is almost fully immersed

at 119879 = 11 and a zero pressure can be seen at the rigid body-air-water interfaceThe fully wet state of the circular cylinderat 119879 = 16 is shown in Figures 6(f) and 7(f) the wave surfacedeformation above the cylinder can be observed and thepressure of the lower part is bigger than that of the upper partof the cylinder

During the transit of a circular cylinder through air-waterinterface an important parameter is the vertical hydrody-namic force on the cylinder also known as the slammingload The slamming coefficient can be defined which is thenondimensional impact force

119862119904 = 1198651205881198812119877 (13)


F

R2

50

40

30

20

10

030 31 32 33 34 35 36

Time (s)

Mesh 1

Mesh 2

Mesh 3

Mesh 4

Figure 5 Hydrodynamic forces on the circular cylinder with different mesh size

(a) (b)

(c) (d)

(e) (f)

Figure 6 Submergence process of a circular cylinder in waves (a) 119879 = 075 (b) 119879 = 0775 (c) 119879 = 084 (d) 119879 = 09 (e) 119879 = 11 (f) 119879 = 16

where 119865 is the vertical hydrodynamic force 119881 is the relativevelocity between the cylinder and wave surface and 119877 is theradius of the cylinder

When 119862119904 is calculated the theoretical relative velocity isused

119881 = V + 119881wave (14)

where119881wave is the velocity ofwater particles at the impact timeof simulation

As shown in Figure 8 the time history of slammingcoefficient during water entry process of the circular cylinderis compared with experimental results by Miao [29] Thefluctuations observed at the beginning part of simulationresults (from V119905119877 = 0 to about V119905119877 = 008) are caused by a

pressure peak in the spray root that covers a smaller area thanthe mesh size

Figure 8 shows that good agreement is obtained betweennumerical results and experimental data at the initial phase(from V119905119877 = 0 to about 014) however from V119905119877 =014 to 1 the slamming coefficient of numerical resultsincrease rapidly and the difference between numerical andexperimental results also increases accordingly Two possibleexplanations for this phenomenon are as follows firstlyin experimental results by Miao it is noteworthy that thevelocity of the cylinder remains constant during water entryprocess while in the present work according to (13) and(14) after slamming the theoretical velocity of water particles119881wave decreases from V119905119877 = 0 to 1 which results in thedecrease of relative velocity between the circular cylinder and


70

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(a)

minus5

0

5

10

15

20

P(1

22)

0 05 1 15 2

x (m)

(b)30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(c)

30

20

10

P(1

22)

0 05 1 15 2

x (m)

(d)

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(e)

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(f)

Figure 7 Pressure distributions on the cylinder (a) 119879 = 075 (b) 119879 = 0775 (c) 119879 = 084 (d) 119879 = 09 (e) 119879 = 11 (f) 119879 = 16


Simulation

Cs line by Miao

0

5

10

15

Cs

10 06 0802 04

tR

Figure 8 Vertical hydrodynamic force on a circular cylinder

water particles and thus the increase of 119862119904 Secondly thewave surface surrounding circular cylinder varies rapidly andremarkably which affects the relative velocity between thecylinder and water particles

35 Influence of the Wave Impact Phase Angle Wave impactphase angles can reflect the relative position between thecircular cylinder and wave surface when impact occursFigure 9 is a diagram of four typical wave impact phaseangles positions marked as 1 2 3 and 4 are phase angles of0∘ 90∘ 180∘ and 270∘ respectively

To investigate the influence of wave impact phase anglespositions marked as 1 2 3 and 4 shown in Figure 7 areapplied The target wave which is simulated in Section 31 isused and the radius 119877 and the constant vertical velocity V ofthe circular cylinder are 1m and 1ms respectively

Two groups of theoretical impact time 119879impact whichrespectively correspond to wave impact phase angles of 0∘90∘ 180∘ and 270∘ are as follows

(1) 30684 s 26184 s 27684 s and 29184 s

(2) 36684 s 32184 s 33684 s and 35184 s

Obviously impact times of the second group are obtainedby those of first group plus period of wave 6 s in this caseFigures 10(a)ndash10(d) reflect time histories of fluid forces on thecircular cylinder during its lowering through wave surfacewith different impact phase angles for the convenience ofcomparison figures of the first group are moved to corre-sponding location of figures of second group with adding6 s in time coordinate named (impact time of the firstgroup + 6 s) Figures 11(a)ndash11(d) show velocity vectors nearthe circular cylinder when the water impact occurs

1

2

3

4

Wave surface

Calm water surface

Figure 9 Wave impact positions

Table 2 Impact time and maximum hydrodynamic force ofdifferent phase angles

Impact phase angle 0∘ 90∘ 180∘ 270∘Time of impact (119879impact) 3664 s 3237 s 3368 s 3513 sTime of 119865max (119879119865) 4118 s 3633 s 3639 s 368 s119879119865 minus 119879impact 456 s 396 s 271 s 167 s

From Figures 10(a)ndash10(d) the following characteristicscan be observed

Firstly the simulated impact time is a little earlier orlater than the theoretical one which is due to calculationand simulation error it is worth noting that no slammingwould occur for case of 180∘ the main reason of which is thatthe relative velocity between cylinder and water particles isvertical downwards at the theoretical impact time which canalso be seen in Figure 11(c)

Secondly the time histories of hydrodynamic forces withthe same impact phase angles are mostly identical whichillustrates that smaller impact time can be applied to savesimulation time as long as the first wave periods are avoided

Thirdly after slamming the hydrodynamic forces grad-ually increase to the maximum value for all four phaseangles which is equal to about 40 kN and this phenomenonindicates the impact wave phase exerts no influence on themaximum hydrodynamic force considering the length of thetime interval between impact time 119879impact and time 119879119865 ofwhich the maximum hydrodynamic force appears it varieswith phase angles which are shown in Table 2 It can be seenthat the interval length decreases with phase angle increasingfrom 0∘ to 270∘

Fourthly after it reaches the maximum value the hydro-dynamic force oscillates near an equilibrium point about31 kN which is equal to the buoyancy of the circular cylinderand the oscillation period is about 6 s which is equal to periodof the target wave besides as simulation time goes on theoscillation amplitude is smaller and smaller and hopefullybecomes zero eventually mainly because of the fact that waveeffect becomes smaller with water depth increasing

As far as the slamming force are concerned cases withphase angle 0∘ have the biggest one which is also the mostdangerous situation for cylinder lowering through wave sur-face mainly due to the biggest relative velocity between thecylinder and water particles which can be observed from


50

40

30

20

10

035 40 45 50 55

F

R2

Time (s)

TＣＧＪ＝Ｎ = 36684 Ｍ

TＣＧＪ＝Ｎ = (30684 + 6) s

(a)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 32184 Ｍ

TＣＧＪ＝Ｎ = (26184 + 6) s

(b)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 33684 Ｍ

TＣＧＪ＝Ｎ = (27684 + 6) Ｍ

(c)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 35184 Ｍ

TＣＧＪ＝Ｎ = (29184 + 6) s

(d)

Figure 10 Time history of hydrodynamic force with different impact phase angles (a) 0∘ (b) 90∘ (c) 180∘ (d) 270∘

Figure 11 Therefore in the following sections only phaseangle 0∘ is used to investigate the effect of other parameters

36 Influence of the Wave Height In order to study the influ-ence of the wave height time histories of hydrodynamic forcewith five different wave heights of regular waves 17m 2m23m 26m and 3m are considered other parameters are asfollows impact phase angle = 0∘ wave period = 6 s constantvelocity of the circular cylinder V = 1ms and cylinder radius119877 = 1m

From Figure 12 the following phenomena can beobserved firstly for all five wave heights hydrodynamicforces have the similar trends and characteristics afterimpactingwithwave surface the forces increase tomaximumvalues and then oscillate near the buoyancy of the circularcylinder with oscillation period equaling the wave periodsand the oscillation amplitude would eventually be zerosecondly both impact forces and maximum hydrodynamicforces increase with an increase of the wave height and theoscillation amplitude of curves have the similar tendency


22

215

21

205

20

195735 74 745 75 755 76 765

Y(m

)

X (m)

minus2 2 6 10 14 18 22Y Velocity

(a)

22

23

215

225

21

205735 74 745 75 755 76 765

Y(m

)

X (m)

minus1minus25minus4 05 2 35Y Velocity

(b)

21

22

205

215

20

735 74 745 75 755 76 765

Y(m

)

X (m)

minus15 minus05minus25minus35 1505Y Velocity

(c)

20

21

195

205

19

185735 74 745 75 755 76 765

Y(m

)

X (m)

minus15 452505 65 85Y Velocity

(d)

Figure 11 Velocity vectors when water impact occurs with different impact phase angles (a) 0∘ (b) 90∘ (c) 180∘ (d) 270∘

thirdly the lengths of time interval between 119879impact and 119879119865are the same for different wave heights with equivalent waveperiods

37 Influence of the Wave Period In order to investigate theinfluence of the wave period time histories of fluid forceswith five different wave periods of regular waves 5 s 6 s 7 s8 s and 9 s are considered which are shown in Figure 13other parameters are as follows impact phase angle = 0∘wave height = 2m constant velocity of the circular cylinderV = 1ms and cylinder radius 119877 = 1m

The following phenomena can be seen from Figure 13firstly both the impact force and the oscillation amplitudeof fluid force curves increase with a decrease of wave periodSecondly hydrodynamic forces oscillate near the buoyancyof the circular cylinder and oscillation periods are equal to

the wave periods Thirdly for cases with periods equaling8 s and 9 s it is worth noting that the first peak force isthe maximum hydrodynamic force which is different fromother cases and first peak forces of all cases increase with anincrease of periods

38 Influence of the Froude Number For the circular cylinderthat is moving through wave surface the Froude number 119865119899is an important dimensionless number which is dependenton vertical velocity and radius of the cylinder To study theeffect of the Froude number on wave impact loads two groupsimulations are performed The first group uses differentradiuses parameters of which are as follows 119877 = 1m 08mand 05m impact phase angle = 0∘ wave height = 2m waveperiod = 6 s constant velocity of the circular cylinder V =1ms which render 119865119899 = 045 036 and 032 The second


50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

H = 17 Ｇ

H = 2Ｇ

H = 23 Ｇ

H = 26 Ｇ

H = 3Ｇ

Figure 12 Fluid forces with different wave heights

50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

T = 6 Ｍ

T = 5 Ｍ

T = 9 Ｍ

T = 8 Ｍ

T = 7 Ｍ

Figure 13 Fluid forces with different wave periods

group uses different velocities and their parameters are asfollows119877=05m impact phase angle = 0∘ wave height = 2mwave period = 6 s constant velocity of the circular cylinder V= 1 2 and 35ms which render 119865119899 = 045 09 and 1575

Figure 14 shows hydrodynamic load on cylinders withdifferent radiuses during the submergence process In thisfigure it can be seen that the nondimensional hydrodynamicload 119865(120588119877V2) increases as the radius of the cylinder becomes

40

30

20

10

0

F

R2

0 6 1812

tR

R = 05 Ｇ (Fn = 045)

R = 08 Ｇ (Fn = 036)

R = 1Ｇ (Fn = 032)

Figure 14 Fluid forces with different cylinder radiuses

30

25

15

20

5

10

0

F

R2

0 6 141210842

tR

= 1ＧＭ (Fn = 045)

= 2ＧＭ (Fn = 09)

= 35ＧＭ (Fn = 1575)

Figure 15 Fluid forces with different vertical velocities

bigger and oscillates near respective buoyancies of cylinderswith different radiuses Figure 15 presents the time historiesof hydrodynamic load 119865(120588119877V2) with different constantvelocities It is seen that fluid forces increase with velocitiesof the cylinder decrease From Figures 14 and 15 it can alsobe seen that the nondimensional slamming forces (the firstpeak force) increase with decreasing Froude number 119865119899 the


main reason is that the smaller the Froude number the moreremarkable the wave exerting influence

4 Conclusions

(1) After the cylinder impacting with wave surface the forcesincrease to some maximum value and then oscillate near thebuoyancy of the cylinder with oscillation period equaling thewave periods and the oscillation amplitude would eventuallybe zero

(2) On one hand cases with wave impact angle 0∘and180∘ respectively have the biggest and smallest impact forcesand cases with other wave impact angles lie within the rangeFor another hand wave impact angle exerts no influence onthe maximum hydrodynamic force during the submergenceprocess

(3) Both the impact force and maximum hydrodynamicforce increase with an increase of wave heights while both ofthem increase with a decrease of wave periods which is dueto the influence of the relative velocity between the cylinderand water particles

(4) The hydrodynamic force is also largely affected bycylinder radius and constant velocity the nondimensionalslamming force decreases with an increase of the Froudenumber

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support ofNatural Science Foundation of Hunan Province of China(Grant no 2017JJ3393) theNational Natural Science Founda-tion of China (no 51305463) and the National Key Researchand Development Plan of China (no 2016YFC0304103)

References

[1] V Karman ldquoThe impact of seaplane floats during landingrdquoational Advisory Committee for Aeronautics pp 1ndash8 1929

[2] H Wanger ldquoTrans phenomena associated with impacts andsliding on liquid surfacesrdquo Journal of Applied Mathematics andMechanics vol 12 no 4 pp 193ndash215 1932

[3] M Greenhow ldquoWater-entry and-exit of a horizontal circularcylinderrdquo Applied Ocean Research vol 10 no 4 pp 191ndash1981988

[4] X M Mei Y M Liu and D K P Yue ldquoOn the water impact ofgeneral two-dimensional sectionsrdquoAppliedOcean Research vol21 no 1 pp 1ndash15 1999

[5] H SunABoundary ElementMethodApplied to Strongly Nonlin-ear Wave-Body Interaction Problems [PhD thesis] NorwegianUniversity of Science and Technology Trondheim Norway2007

[6] X Zhu Application of the CIP Method to Strongly NonlinearWave-Body Interaction Problems [PhD thesis] NorwegianUni-versity of Science and Technology Trondheim Norway 2006

[7] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[8] K Zhang K Yan X-S Chu and G-Y Chen ldquoNumericalsimulation of the water-entry of body based on the LatticeBoltzmann methodrdquo Journal of Hydrodynamics vol 22 no 5pp 829ndash833 2010

[9] J Vandamme Q Zou and D E Reeve ldquoModeling floatingobject entry and exit using smoothed particle hydrodynamicsrdquoJournal of Waterway Port Coastal and Ocean Engineering vol137 no 5 pp 213ndash224 2011

[10] A Skillen S Lind P K Stansby and B D Rogers ldquoIncompress-ible smoothed particle hydrodynamics (SPH) with reducedtemporal noise and generalised Fickian smoothing applied tobody-water slam and efficient wave-body interactionrdquo Com-puter Methods Applied Mechanics and Engineering vol 265 pp163ndash173 2013

[11] H B Gu L Qian D M Causon C G Mingham and P LinldquoNumerical simulation of water impact of solid bodies withvertical and oblique entriesrdquoOceanEngineering vol 75 pp 128ndash137 2014

[12] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers and Fluids vol 140 pp19ndash38 2016

[13] W Peng and Q Wei ldquoSolving 2-D water entry problems witha CIP method and a parallel computing algorithmrdquo MarineSystems Ocean Technology vol 11 no 1-2 p 1 2016

[14] A Iranmanesh andM Passandideh-Fard ldquoA three-dimensionalnumerical approach on water entry of a horizontal circularcylinder using the volume of fluid techniquerdquo Ocean Engineer-ing vol 130 pp 557ndash566 2017

[15] P Nair and G Tomar ldquoA study of energy transfer duringwater entry of solids using incompressible SPH simulationsrdquoSadhana vol 42 no 4 pp 517ndash531 2017

[16] F Aristodemo G Tripepi D D Meringolo and P Veltri ldquoSoli-tary wave-induced forces on horizontal circular cylinders lab-oratory experiments and SPH simulationsrdquoCoastal EngineeringJournal vol 129 pp 17ndash35 2017

[17] C Mnasri Z Hafsia M Omri and K Maalel ldquoA moving gridmodel for simulation of free surface behavior induced byhorizontal cylinders exit and entryrdquo Engineering Applications ofComputational Fluid Mechanics vol 4 no 2 pp 260ndash275 2014

[18] Y Chen Y Gao and Q Liu ldquoNumerical simulation of water-entry in a horizontal circular cylinder using the volume of fluid(VOF) methodrdquo Journal of Harbin Engineering University vol32 no 11 pp 1439ndash1442 2011

[19] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[20] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[21] E Larsen Impact Loads on Circular Cylinders Norwegian Uni-versity of Science and Technology Trondheim Norway 2013

[22] B E Launder and D B Spalding Lectures in MathematicalModels of Turbulence Academic Press London UK 1972

[23] ANSYS ANSYS Fluent Userrsquos Guide 2013[24] ANSYS ANSYS Fluent Theory Guide 2013


[25] M Greenhow and S Moyo ldquoWater entry and exit of horizontalcircular cylindersrdquo Philosophical Transactions of the RoyalSociety of London Series A Mathematical Physical Sciences andEngineering vol 355 no 1724 pp 551ndash563 1997

[26] P Lin ldquoA fixed-grid model for simulation of a moving body infree surface flowsrdquo Computers amp Fluids vol 36 no 3 pp 549ndash561 2007

[27] P A Tyvand and T Miloh ldquoFree-surface flow due to impulsivemotion of a submerged circular cylinderrdquo Journal of FluidMechanics vol 286 pp 67ndash101 1995

[28] T Campbell and P Weynberg ldquoMeasurement of parametersaffecting slamming - final reportrdquo Tech Rep 440 WolfsonMarine Craft Unit University of Southampton 1980

[29] G MiaoHydrodynamic forces and dynamic responses of circularcylinders in wave zones [PhD thesis] Dept of Marine Hydro-dynamics Trondheim Norway 1989

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Navier-Stokes solver [12] Peng and Wei simulated the waterentry of a circular cylinder based on a CIP method withthe parallel algorithm [13] Iranmanesh and Passandideh-Fard numerically investigated water entry of a horizontal cir-cular cylinder for low Froude numbers by the combinationof the fast-fictitious-domain method and the volume-of-fluid (VOF) technique [14] NAIR and TOMAR employedan incompressible SmoothedParticleHydrodynamics (ISPH)method to simulate water entry of 2D circular cylinders[15] Aristodemo et al performed numerical study on wave-induced forces of submerged circular cylinders by SPHmethod [16] Advances in computer technology and CFDhave made it possible to use commercial CFD codes to solvewave impact problem For example Mnasri et al used Fluentcode with a moving grid to analyze the free surface evolutioninduced by one or two moving cylinders [17] Chen et alstudied water entry of a horizontal cylinder based on VOFmethod by Fluent [18] Ghadimi et al utilized FLOW-3Dcodeto study thewater entry of a circular cylinder and conducted acomparison between the linear and nonlinear solutions [19]Tassin et al investigated water impact problem of a body withtime-varying shape by CFD code OpenFOAM [20] Larsenused the CFD code STAR-CCM+ to calculate impact loadson circular cylinders during water entry [21]

This work is devoted to investigate the complex inter-action between a circular cylinder and waves during thesubmergence process Firstly both governing equations andboundary conditions are outlined to study the interactionbetween wave and cylinders a numerical wave tank isestablished and numerical wave generation and absorptionmethod are presented Secondly snapshots of the simulationof calm water entry of a circular cylinder are compared withthose of experiments by previous researchesThen numericalsimulations of a circular cylinder impact with waves areconducted and free surface profiles of different time instantsof submergence process are shown Finally slamming loadson the circular cylinder are computed and influence of waveparameters is discussed

2 The Numerical Method

21 Governing Equations In this work governing equationsfor the CFD calculations are RANS equations for homoge-neous incompressible fluid flows and they are written asfollows

120597119906119894120597119909119894 = 0 (119894 = 1 2) 120597 (120588119906119894)120597119905 + 120597 (120588119906119894119906119895)120597119909119894 = minus 120597119901120597119909119894

+ 120597120597119909119895 [120583(120597119906119894120597119909119895 +

120597119906119895120597119909119894 )]

+ 120597120597119909119895 (minus12058811990610158401198941199061015840119895) + 120588119891119894

(1)

where 119894 119895 = 1 2 for two-dimensional flows 119906119894 is the averagevelocity of 119894th coordinate axis 120588 is the density119901 is the averagepressure120583 is the dynamic viscous coefficient119891119894 is the averagebody force minus12058811990610158401198941199061015840119895 is the Reynolds stress120597 (120588119896)120597119905 +

120597 (120588119896119906119895)120597119909119894 = 120597120597119909119895 [(120583 +

120583119905120590119896)120597119896120597119909119895] + 120588119866119896

minus 120588120576120597 (120588120576)120597119905 +

120597 (120588120576119906119895)120597119909119894 = 120597120597119909119895 [(120583 +

120583119905120590120576)120597119896120597119909119895]

+ 120576119896 (1198621205761120588119866119896 minus 1198621205762120588120576)

120583119905 = 1198621205831205881198962

120576

119866119896 = 120583119905120588 (120597119906119894120597119909119895 +

120597119906119895120597119909119894 )

120597119906119894120597119909119895

(2)

where 119896 is the turbulence kinetic energy 120576 is the dissipationrate of turbulence kinetic energy 120583119905 is the eddy viscosity 119866119896is the turbulent energy production

The empirical coefficients in (2) are given as follows [22]

119862120583 = 0091198621205761 = 1441198621205762 = 192120590119896 = 10120590120576 = 13

(3)

The complex free surface is tracked by the VOF methodand to accomplish the capturing of the interface between theair phase and the water phase a continuity equation for thefraction of volume of water is solved which has the followingform

120597119886119908120597119905 + 119906119894120597 (119886119908)120597119909119894 = 0 (4)

The sum of the volume fraction of water and air is givenby

119886119908 + 119886119886 = 1 (5)

where 119886119908 and 119886119886 are the volume fraction of water and air ineach cell respectively

22 Boundary Conditions To solve governing equations it isnecessary to specify appropriate boundary conditions at allboundaries of the domain The boundary conditions whichneed to be satisfied are as follows (1) the kinematic anddynamic free surface conditions at the free surface and (2)


z

x

Dampingzone

Inputboundary

L

Wavegeneration

zone

Workingzone

H

Circular cylinder

d

Waves

L0 Ld

Ω(t)

Bottom Γb

Wall Γw

Free surface Γf

ΓI





119883(119905) = 11988302 sin120596119905 (6)


120578 (119909 119905) = 11988302 (4 sinh2119896119889

2119896119889 + sinh 2119896119889 cos (119896119909 minus 120596119905)

+ infinsum119899=1

4 sin2 (119896119899119889)2119896119899119889 + sin (2119896119899119889)119890minus119896119899119909 sin120596119905)

(7)



120578 (119909 119905) = 119883024 sinh2119896119889


(8)


119867 = 4 sinh2 (119896119889)2119896119889 + sinh (2119896119889)1198830 (9)


119878119894 = minus(120583120572V119894 + 119862212120588 1003816100381610038161003816V1198941003816100381610038161003816 V119894) (10)











Δ119905 le 119862Δ119909V119891 (11)



1120572 (119909) = 1205720 (

119909 minus 119909119904119909119890 minus 119909119904) (12)






100

0

minus100

A

()

0 20 40 60 80 100

Time (s)

(a)

100

0

minus100

A

()

0 20 40 60 80 100

Time (s)

(b)

50

0

minus50

A

()

0 20 40 60 80 100

Time (s)

(c)

5

0

minus5

A

()

0 20 40 60 80 100

Time (s)

(d)















1

0

minus1

minus2


xd

yd

(a) 119879 = 00

1

0

minus1

minus2


xd

yd

(b) 119879 = 04

1

0

minus1

minus2


xd

yd

(c) 119879 = 10

1

0

minus1

minus2


xd

yd

(d) 119879 = 20





119862119904 = 1198651205881198812119877 (13)


F

R2

50

40

30

20

10

030 31 32 33 34 35 36

Time (s)

Mesh 1

Mesh 2

Mesh 3

Mesh 4


(a) (b)

(c) (d)

(e) (f)




119881 = V + 119881wave (14)






70

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(a)

minus5

0

5

10

15

20

P(1

22)

0 05 1 15 2

x (m)

(b)30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(c)

30

20

10

P(1

22)

0 05 1 15 2

x (m)

(d)

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(e)

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(f)



Simulation

Cs line by Miao

0

5

10

15

Cs

10 06 0802 04

tR






(1) 30684 s 26184 s 27684 s and 29184 s

(2) 36684 s 32184 s 33684 s and 35184 s


1

2

3

4

Wave surface

Calm water surface











50

40

30

20

10

035 40 45 50 55

F

R2

Time (s)

TＣＧＪ＝Ｎ = 36684 Ｍ

TＣＧＪ＝Ｎ = (30684 + 6) s

(a)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 32184 Ｍ

TＣＧＪ＝Ｎ = (26184 + 6) s

(b)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 33684 Ｍ

TＣＧＪ＝Ｎ = (27684 + 6) Ｍ

(c)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 35184 Ｍ

TＣＧＪ＝Ｎ = (29184 + 6) s

(d)






22

215

21

205

20

195735 74 745 75 755 76 765

Y(m

)

X (m)


(a)

22

23

215

225

21

205735 74 745 75 755 76 765

Y(m

)

X (m)


(b)

21

22

205

215

20

735 74 745 75 755 76 765

Y(m

)

X (m)


(c)

20

21

195

205

19

185735 74 745 75 755 76 765

Y(m

)

X (m)


(d)








50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

H = 17 Ｇ

H = 2Ｇ

H = 23 Ｇ

H = 26 Ｇ

H = 3Ｇ


50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

T = 6 Ｍ

T = 5 Ｍ

T = 9 Ｍ

T = 8 Ｍ

T = 7 Ｍ




40

30

20

10

0

F

R2

0 6 1812

tR

R = 05 Ｇ (Fn = 045)

R = 08 Ｇ (Fn = 036)

R = 1Ｇ (Fn = 032)


30

25

15

20

5

10

0

F

R2

0 6 141210842

tR

= 1ＧＭ (Fn = 045)

= 2ＧＭ (Fn = 09)

= 35ＧＭ (Fn = 1575)





4 Conclusions







Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










z

x

Dampingzone

Inputboundary

L

Wavegeneration

zone

Workingzone

H

Circular cylinder

d

Waves

L0 Ld

Ω(t)

Bottom Γb

Wall Γw

Free surface Γf

ΓI





119883(119905) = 11988302 sin120596119905 (6)


120578 (119909 119905) = 11988302 (4 sinh2119896119889

2119896119889 + sinh 2119896119889 cos (119896119909 minus 120596119905)

+ infinsum119899=1

4 sin2 (119896119899119889)2119896119899119889 + sin (2119896119899119889)119890minus119896119899119909 sin120596119905)

(7)



120578 (119909 119905) = 119883024 sinh2119896119889


(8)


119867 = 4 sinh2 (119896119889)2119896119889 + sinh (2119896119889)1198830 (9)


119878119894 = minus(120583120572V119894 + 119862212120588 1003816100381610038161003816V1198941003816100381610038161003816 V119894) (10)











Δ119905 le 119862Δ119909V119891 (11)



1120572 (119909) = 1205720 (

119909 minus 119909119904119909119890 minus 119909119904) (12)






100

0

minus100

A

()

0 20 40 60 80 100

Time (s)

(a)

100

0

minus100

A

()

0 20 40 60 80 100

Time (s)

(b)

50

0

minus50

A

()

0 20 40 60 80 100

Time (s)

(c)

5

0

minus5

A

()

0 20 40 60 80 100

Time (s)

(d)















1

0

minus1

minus2


xd

yd

(a) 119879 = 00

1

0

minus1

minus2


xd

yd

(b) 119879 = 04

1

0

minus1

minus2


xd

yd

(c) 119879 = 10

1

0

minus1

minus2


xd

yd

(d) 119879 = 20





119862119904 = 1198651205881198812119877 (13)


F

R2

50

40

30

20

10

030 31 32 33 34 35 36

Time (s)

Mesh 1

Mesh 2

Mesh 3

Mesh 4


(a) (b)

(c) (d)

(e) (f)




119881 = V + 119881wave (14)






70

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(a)

minus5

0

5

10

15

20

P(1

22)

0 05 1 15 2

x (m)

(b)30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(c)

30

20

10

P(1

22)

0 05 1 15 2

x (m)

(d)

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(e)

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(f)



Simulation

Cs line by Miao

0

5

10

15

Cs

10 06 0802 04

tR






(1) 30684 s 26184 s 27684 s and 29184 s

(2) 36684 s 32184 s 33684 s and 35184 s


1

2

3

4

Wave surface

Calm water surface











50

40

30

20

10

035 40 45 50 55

F

R2

Time (s)

TＣＧＪ＝Ｎ = 36684 Ｍ

TＣＧＪ＝Ｎ = (30684 + 6) s

(a)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 32184 Ｍ

TＣＧＪ＝Ｎ = (26184 + 6) s

(b)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 33684 Ｍ

TＣＧＪ＝Ｎ = (27684 + 6) Ｍ

(c)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 35184 Ｍ

TＣＧＪ＝Ｎ = (29184 + 6) s

(d)






22

215

21

205

20

195735 74 745 75 755 76 765

Y(m

)

X (m)


(a)

22

23

215

225

21

205735 74 745 75 755 76 765

Y(m

)

X (m)


(b)

21

22

205

215

20

735 74 745 75 755 76 765

Y(m

)

X (m)


(c)

20

21

195

205

19

185735 74 745 75 755 76 765

Y(m

)

X (m)


(d)








50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

H = 17 Ｇ

H = 2Ｇ

H = 23 Ｇ

H = 26 Ｇ

H = 3Ｇ


50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

T = 6 Ｍ

T = 5 Ｍ

T = 9 Ｍ

T = 8 Ｍ

T = 7 Ｍ




40

30

20

10

0

F

R2

0 6 1812

tR

R = 05 Ｇ (Fn = 045)

R = 08 Ｇ (Fn = 036)

R = 1Ｇ (Fn = 032)


30

25

15

20

5

10

0

F

R2

0 6 141210842

tR

= 1ＧＭ (Fn = 045)

= 2ＧＭ (Fn = 09)

= 35ＧＭ (Fn = 1575)





4 Conclusions







Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














Δ119905 le 119862Δ119909V119891 (11)



1120572 (119909) = 1205720 (

119909 minus 119909119904119909119890 minus 119909119904) (12)






100

0

minus100

A

()

0 20 40 60 80 100

Time (s)

(a)

100

0

minus100

A

()

0 20 40 60 80 100

Time (s)

(b)

50

0

minus50

A

()

0 20 40 60 80 100

Time (s)

(c)

5

0

minus5

A

()

0 20 40 60 80 100

Time (s)

(d)















1

0

minus1

minus2


xd

yd

(a) 119879 = 00

1

0

minus1

minus2


xd

yd

(b) 119879 = 04

1

0

minus1

minus2


xd

yd

(c) 119879 = 10

1

0

minus1

minus2


xd

yd

(d) 119879 = 20





119862119904 = 1198651205881198812119877 (13)


F

R2

50

40

30

20

10

030 31 32 33 34 35 36

Time (s)

Mesh 1

Mesh 2

Mesh 3

Mesh 4


(a) (b)

(c) (d)

(e) (f)




119881 = V + 119881wave (14)






70

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(a)

minus5

0

5

10

15

20

P(1

22)

0 05 1 15 2

x (m)

(b)30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(c)

30

20

10

P(1

22)

0 05 1 15 2

x (m)

(d)

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(e)

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(f)



Simulation

Cs line by Miao

0

5

10

15

Cs

10 06 0802 04

tR






(1) 30684 s 26184 s 27684 s and 29184 s

(2) 36684 s 32184 s 33684 s and 35184 s


1

2

3

4

Wave surface

Calm water surface











50

40

30

20

10

035 40 45 50 55

F

R2

Time (s)

TＣＧＪ＝Ｎ = 36684 Ｍ

TＣＧＪ＝Ｎ = (30684 + 6) s

(a)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 32184 Ｍ

TＣＧＪ＝Ｎ = (26184 + 6) s

(b)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 33684 Ｍ

TＣＧＪ＝Ｎ = (27684 + 6) Ｍ

(c)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 35184 Ｍ

TＣＧＪ＝Ｎ = (29184 + 6) s

(d)






22

215

21

205

20

195735 74 745 75 755 76 765

Y(m

)

X (m)


(a)

22

23

215

225

21

205735 74 745 75 755 76 765

Y(m

)

X (m)


(b)

21

22

205

215

20

735 74 745 75 755 76 765

Y(m

)

X (m)


(c)

20

21

195

205

19

185735 74 745 75 755 76 765

Y(m

)

X (m)


(d)








50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

H = 17 Ｇ

H = 2Ｇ

H = 23 Ｇ

H = 26 Ｇ

H = 3Ｇ


50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

T = 6 Ｍ

T = 5 Ｍ

T = 9 Ｍ

T = 8 Ｍ

T = 7 Ｍ




40

30

20

10

0

F

R2

0 6 1812

tR

R = 05 Ｇ (Fn = 045)

R = 08 Ｇ (Fn = 036)

R = 1Ｇ (Fn = 032)


30

25

15

20

5

10

0

F

R2

0 6 141210842

tR

= 1ＧＭ (Fn = 045)

= 2ＧＭ (Fn = 09)

= 35ＧＭ (Fn = 1575)





4 Conclusions







Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










100

0

minus100

A

()

0 20 40 60 80 100

Time (s)

(a)

100

0

minus100

A

()

0 20 40 60 80 100

Time (s)

(b)

50

0

minus50

A

()

0 20 40 60 80 100

Time (s)

(c)

5

0

minus5

A

()

0 20 40 60 80 100

Time (s)

(d)















1

0

minus1

minus2


xd

yd

(a) 119879 = 00

1

0

minus1

minus2


xd

yd

(b) 119879 = 04

1

0

minus1

minus2


xd

yd

(c) 119879 = 10

1

0

minus1

minus2


xd

yd

(d) 119879 = 20





119862119904 = 1198651205881198812119877 (13)


F

R2

50

40

30

20

10

030 31 32 33 34 35 36

Time (s)

Mesh 1

Mesh 2

Mesh 3

Mesh 4


(a) (b)

(c) (d)

(e) (f)




119881 = V + 119881wave (14)






70

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(a)

minus5

0

5

10

15

20

P(1

22)

0 05 1 15 2

x (m)

(b)30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(c)

30

20

10

P(1

22)

0 05 1 15 2

x (m)

(d)

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(e)

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(f)



Simulation

Cs line by Miao

0

5

10

15

Cs

10 06 0802 04

tR






(1) 30684 s 26184 s 27684 s and 29184 s

(2) 36684 s 32184 s 33684 s and 35184 s


1

2

3

4

Wave surface

Calm water surface











50

40

30

20

10

035 40 45 50 55

F

R2

Time (s)

TＣＧＪ＝Ｎ = 36684 Ｍ

TＣＧＪ＝Ｎ = (30684 + 6) s

(a)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 32184 Ｍ

TＣＧＪ＝Ｎ = (26184 + 6) s

(b)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 33684 Ｍ

TＣＧＪ＝Ｎ = (27684 + 6) Ｍ

(c)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 35184 Ｍ

TＣＧＪ＝Ｎ = (29184 + 6) s

(d)






22

215

21

205

20

195735 74 745 75 755 76 765

Y(m

)

X (m)


(a)

22

23

215

225

21

205735 74 745 75 755 76 765

Y(m

)

X (m)


(b)

21

22

205

215

20

735 74 745 75 755 76 765

Y(m

)

X (m)


(c)

20

21

195

205

19

185735 74 745 75 755 76 765

Y(m

)

X (m)


(d)








50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

H = 17 Ｇ

H = 2Ｇ

H = 23 Ｇ

H = 26 Ｇ

H = 3Ｇ


50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

T = 6 Ｍ

T = 5 Ｍ

T = 9 Ｍ

T = 8 Ｍ

T = 7 Ｍ




40

30

20

10

0

F

R2

0 6 1812

tR

R = 05 Ｇ (Fn = 045)

R = 08 Ｇ (Fn = 036)

R = 1Ｇ (Fn = 032)


30

25

15

20

5

10

0

F

R2

0 6 141210842

tR

= 1ＧＭ (Fn = 045)

= 2ＧＭ (Fn = 09)

= 35ＧＭ (Fn = 1575)





4 Conclusions







Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

0

minus1

minus2


xd

yd

(a) 119879 = 00

1

0

minus1

minus2


xd

yd

(b) 119879 = 04

1

0

minus1

minus2


xd

yd

(c) 119879 = 10

1

0

minus1

minus2


xd

yd

(d) 119879 = 20





119862119904 = 1198651205881198812119877 (13)


F

R2

50

40

30

20

10

030 31 32 33 34 35 36

Time (s)

Mesh 1

Mesh 2

Mesh 3

Mesh 4


(a) (b)

(c) (d)

(e) (f)




119881 = V + 119881wave (14)






70

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(a)

minus5

0

5

10

15

20

P(1

22)

0 05 1 15 2

x (m)

(b)30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(c)

30

20

10

P(1

22)

0 05 1 15 2

x (m)

(d)

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(e)

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(f)



Simulation

Cs line by Miao

0

5

10

15

Cs

10 06 0802 04

tR






(1) 30684 s 26184 s 27684 s and 29184 s

(2) 36684 s 32184 s 33684 s and 35184 s


1

2

3

4

Wave surface

Calm water surface











50

40

30

20

10

035 40 45 50 55

F

R2

Time (s)

TＣＧＪ＝Ｎ = 36684 Ｍ

TＣＧＪ＝Ｎ = (30684 + 6) s

(a)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 32184 Ｍ

TＣＧＪ＝Ｎ = (26184 + 6) s

(b)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 33684 Ｍ

TＣＧＪ＝Ｎ = (27684 + 6) Ｍ

(c)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 35184 Ｍ

TＣＧＪ＝Ｎ = (29184 + 6) s

(d)






22

215

21

205

20

195735 74 745 75 755 76 765

Y(m

)

X (m)


(a)

22

23

215

225

21

205735 74 745 75 755 76 765

Y(m

)

X (m)


(b)

21

22

205

215

20

735 74 745 75 755 76 765

Y(m

)

X (m)


(c)

20

21

195

205

19

185735 74 745 75 755 76 765

Y(m

)

X (m)


(d)








50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

H = 17 Ｇ

H = 2Ｇ

H = 23 Ｇ

H = 26 Ｇ

H = 3Ｇ


50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

T = 6 Ｍ

T = 5 Ｍ

T = 9 Ｍ

T = 8 Ｍ

T = 7 Ｍ




40

30

20

10

0

F

R2

0 6 1812

tR

R = 05 Ｇ (Fn = 045)

R = 08 Ｇ (Fn = 036)

R = 1Ｇ (Fn = 032)


30

25

15

20

5

10

0

F

R2

0 6 141210842

tR

= 1ＧＭ (Fn = 045)

= 2ＧＭ (Fn = 09)

= 35ＧＭ (Fn = 1575)





4 Conclusions







Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










F

R2

50

40

30

20

10

030 31 32 33 34 35 36

Time (s)

Mesh 1

Mesh 2

Mesh 3

Mesh 4


(a) (b)

(c) (d)

(e) (f)




119881 = V + 119881wave (14)






70

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(a)

minus5

0

5

10

15

20

P(1

22)

0 05 1 15 2

x (m)

(b)30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(c)

30

20

10

P(1

22)

0 05 1 15 2

x (m)

(d)

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(e)

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(f)



Simulation

Cs line by Miao

0

5

10

15

Cs

10 06 0802 04

tR






(1) 30684 s 26184 s 27684 s and 29184 s

(2) 36684 s 32184 s 33684 s and 35184 s


1

2

3

4

Wave surface

Calm water surface











50

40

30

20

10

035 40 45 50 55

F

R2

Time (s)

TＣＧＪ＝Ｎ = 36684 Ｍ

TＣＧＪ＝Ｎ = (30684 + 6) s

(a)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 32184 Ｍ

TＣＧＪ＝Ｎ = (26184 + 6) s

(b)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 33684 Ｍ

TＣＧＪ＝Ｎ = (27684 + 6) Ｍ

(c)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 35184 Ｍ

TＣＧＪ＝Ｎ = (29184 + 6) s

(d)






22

215

21

205

20

195735 74 745 75 755 76 765

Y(m

)

X (m)


(a)

22

23

215

225

21

205735 74 745 75 755 76 765

Y(m

)

X (m)


(b)

21

22

205

215

20

735 74 745 75 755 76 765

Y(m

)

X (m)


(c)

20

21

195

205

19

185735 74 745 75 755 76 765

Y(m

)

X (m)


(d)








50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

H = 17 Ｇ

H = 2Ｇ

H = 23 Ｇ

H = 26 Ｇ

H = 3Ｇ


50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

T = 6 Ｍ

T = 5 Ｍ

T = 9 Ｍ

T = 8 Ｍ

T = 7 Ｍ




40

30

20

10

0

F

R2

0 6 1812

tR

R = 05 Ｇ (Fn = 045)

R = 08 Ｇ (Fn = 036)

R = 1Ｇ (Fn = 032)


30

25

15

20

5

10

0

F

R2

0 6 141210842

tR

= 1ＧＭ (Fn = 045)

= 2ＧＭ (Fn = 09)

= 35ＧＭ (Fn = 1575)





4 Conclusions







Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










70

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(a)

minus5

0

5

10

15

20

P(1

22)

0 05 1 15 2

x (m)

(b)30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(c)

30

20

10

P(1

22)

0 05 1 15 2

x (m)

(d)

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(e)

60

50

40

30

20

10

0

P(1

22)

0 05 1 15 2

x (m)

(f)



Simulation

Cs line by Miao

0

5

10

15

Cs

10 06 0802 04

tR






(1) 30684 s 26184 s 27684 s and 29184 s

(2) 36684 s 32184 s 33684 s and 35184 s


1

2

3

4

Wave surface

Calm water surface











50

40

30

20

10

035 40 45 50 55

F

R2

Time (s)

TＣＧＪ＝Ｎ = 36684 Ｍ

TＣＧＪ＝Ｎ = (30684 + 6) s

(a)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 32184 Ｍ

TＣＧＪ＝Ｎ = (26184 + 6) s

(b)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 33684 Ｍ

TＣＧＪ＝Ｎ = (27684 + 6) Ｍ

(c)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 35184 Ｍ

TＣＧＪ＝Ｎ = (29184 + 6) s

(d)






22

215

21

205

20

195735 74 745 75 755 76 765

Y(m

)

X (m)


(a)

22

23

215

225

21

205735 74 745 75 755 76 765

Y(m

)

X (m)


(b)

21

22

205

215

20

735 74 745 75 755 76 765

Y(m

)

X (m)


(c)

20

21

195

205

19

185735 74 745 75 755 76 765

Y(m

)

X (m)


(d)








50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

H = 17 Ｇ

H = 2Ｇ

H = 23 Ｇ

H = 26 Ｇ

H = 3Ｇ


50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

T = 6 Ｍ

T = 5 Ｍ

T = 9 Ｍ

T = 8 Ｍ

T = 7 Ｍ




40

30

20

10

0

F

R2

0 6 1812

tR

R = 05 Ｇ (Fn = 045)

R = 08 Ｇ (Fn = 036)

R = 1Ｇ (Fn = 032)


30

25

15

20

5

10

0

F

R2

0 6 141210842

tR

= 1ＧＭ (Fn = 045)

= 2ＧＭ (Fn = 09)

= 35ＧＭ (Fn = 1575)





4 Conclusions







Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Simulation

Cs line by Miao

0

5

10

15

Cs

10 06 0802 04

tR






(1) 30684 s 26184 s 27684 s and 29184 s

(2) 36684 s 32184 s 33684 s and 35184 s


1

2

3

4

Wave surface

Calm water surface











50

40

30

20

10

035 40 45 50 55

F

R2

Time (s)

TＣＧＪ＝Ｎ = 36684 Ｍ

TＣＧＪ＝Ｎ = (30684 + 6) s

(a)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 32184 Ｍ

TＣＧＪ＝Ｎ = (26184 + 6) s

(b)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 33684 Ｍ

TＣＧＪ＝Ｎ = (27684 + 6) Ｍ

(c)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 35184 Ｍ

TＣＧＪ＝Ｎ = (29184 + 6) s

(d)






22

215

21

205

20

195735 74 745 75 755 76 765

Y(m

)

X (m)


(a)

22

23

215

225

21

205735 74 745 75 755 76 765

Y(m

)

X (m)


(b)

21

22

205

215

20

735 74 745 75 755 76 765

Y(m

)

X (m)


(c)

20

21

195

205

19

185735 74 745 75 755 76 765

Y(m

)

X (m)


(d)








50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

H = 17 Ｇ

H = 2Ｇ

H = 23 Ｇ

H = 26 Ｇ

H = 3Ｇ


50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

T = 6 Ｍ

T = 5 Ｍ

T = 9 Ｍ

T = 8 Ｍ

T = 7 Ｍ




40

30

20

10

0

F

R2

0 6 1812

tR

R = 05 Ｇ (Fn = 045)

R = 08 Ｇ (Fn = 036)

R = 1Ｇ (Fn = 032)


30

25

15

20

5

10

0

F

R2

0 6 141210842

tR

= 1ＧＭ (Fn = 045)

= 2ＧＭ (Fn = 09)

= 35ＧＭ (Fn = 1575)





4 Conclusions







Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










50

40

30

20

10

035 40 45 50 55

F

R2

Time (s)

TＣＧＪ＝Ｎ = 36684 Ｍ

TＣＧＪ＝Ｎ = (30684 + 6) s

(a)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 32184 Ｍ

TＣＧＪ＝Ｎ = (26184 + 6) s

(b)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 33684 Ｍ

TＣＧＪ＝Ｎ = (27684 + 6) Ｍ

(c)

50

40

30

20

10

030 35 40 45 50

F

R2

Time (s)

TＣＧＪ＝Ｎ = 35184 Ｍ

TＣＧＪ＝Ｎ = (29184 + 6) s

(d)






22

215

21

205

20

195735 74 745 75 755 76 765

Y(m

)

X (m)


(a)

22

23

215

225

21

205735 74 745 75 755 76 765

Y(m

)

X (m)


(b)

21

22

205

215

20

735 74 745 75 755 76 765

Y(m

)

X (m)


(c)

20

21

195

205

19

185735 74 745 75 755 76 765

Y(m

)

X (m)


(d)








50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

H = 17 Ｇ

H = 2Ｇ

H = 23 Ｇ

H = 26 Ｇ

H = 3Ｇ


50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

T = 6 Ｍ

T = 5 Ｍ

T = 9 Ｍ

T = 8 Ｍ

T = 7 Ｍ




40

30

20

10

0

F

R2

0 6 1812

tR

R = 05 Ｇ (Fn = 045)

R = 08 Ｇ (Fn = 036)

R = 1Ｇ (Fn = 032)


30

25

15

20

5

10

0

F

R2

0 6 141210842

tR

= 1ＧＭ (Fn = 045)

= 2ＧＭ (Fn = 09)

= 35ＧＭ (Fn = 1575)





4 Conclusions







Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










22

215

21

205

20

195735 74 745 75 755 76 765

Y(m

)

X (m)


(a)

22

23

215

225

21

205735 74 745 75 755 76 765

Y(m

)

X (m)


(b)

21

22

205

215

20

735 74 745 75 755 76 765

Y(m

)

X (m)


(c)

20

21

195

205

19

185735 74 745 75 755 76 765

Y(m

)

X (m)


(d)








50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

H = 17 Ｇ

H = 2Ｇ

H = 23 Ｇ

H = 26 Ｇ

H = 3Ｇ


50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

T = 6 Ｍ

T = 5 Ｍ

T = 9 Ｍ

T = 8 Ｍ

T = 7 Ｍ




40

30

20

10

0

F

R2

0 6 1812

tR

R = 05 Ｇ (Fn = 045)

R = 08 Ｇ (Fn = 036)

R = 1Ｇ (Fn = 032)


30

25

15

20

5

10

0

F

R2

0 6 141210842

tR

= 1ＧＭ (Fn = 045)

= 2ＧＭ (Fn = 09)

= 35ＧＭ (Fn = 1575)





4 Conclusions







Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

H = 17 Ｇ

H = 2Ｇ

H = 23 Ｇ

H = 26 Ｇ

H = 3Ｇ


50

40

30

20

10

0

F

R2

0 2 4 6 8 10 12

tR

T = 6 Ｍ

T = 5 Ｍ

T = 9 Ｍ

T = 8 Ｍ

T = 7 Ｍ




40

30

20

10

0

F

R2

0 6 1812

tR

R = 05 Ｇ (Fn = 045)

R = 08 Ｇ (Fn = 036)

R = 1Ｇ (Fn = 032)


30

25

15

20

5

10

0

F

R2

0 6 141210842

tR

= 1ＧＭ (Fn = 045)

= 2ＧＭ (Fn = 09)

= 35ＧＭ (Fn = 1575)





4 Conclusions







Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











4 Conclusions







Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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