Research Article Finite Element Simulation of Dynamic...

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2013 Article ID 252760 13 pageshttpdxdoiorg1011552013252760

Research ArticleFinite Element Simulation of Dynamic Stability of PlaneFree-Surface of a Liquid under Vertical Excitation

Siva Srinivas Kolukula and P Chellapandi

Structural Mechanics Laboratory Indira Gandhi Center for Atomic Research Kalpakkam Tamil Nadu 603102 India

Correspondence should be addressed to Siva Srinivas Kolukula allwayzitzmegmailcom

Received 8 July 2013 Accepted 15 September 2013

Academic Editor Joseph Virgone

Copyright copy 2013 S S Kolukula and P Chellapandi This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

When partially filled liquid containers are excited vertically the plane free-surface of the liquid can be stable or unstable dependingon the amplitude and frequency of the external excitation For some combinations of amplitude and frequency the free-surfaceundergoes unbounded motion leading to instability called parametric instability or parametric resonance and for few othercombinations the free-surface undergoes bounded stable motion In parametric resonance a small initial perturbation on thefree-surface can build up unboundedly even for small external excitation if the excitation acts on the tank for sufficiently longtime In this paper the stability of the plane free-surface is investigated by numerical simulation Stability chart for the governingMathieu equation is plotted analytically using linear equations Applying fully nonlinear finite element method based on nonlinearpotential theory the response of the plane free-surface is simulated for various cases

1 Introduction

In many engineering applications it is of great interest tostudy the oscillatory motion of a mechanical system underexternal excitations In all of the vibrating systems it is essen-tial to know the response of the system and the natureof the oscillations It is very important to get informationabout the different resonance cases of the system to avoidthe harmful ones in its design Under external periodic exci-tations the system may undergo two kinds of oscillationsforced oscillations and parametric oscillations Forced oscil-lations correspond to the oscillatory response of the systemin the direction of external excitation and the system under-goes resonance when external excitation is equal to naturalfrequency of the system In resonance systems amplitudeincreases linearly Parametric oscillations refer to an oscil-latory motion in a mechanical system due to time-varyingchanges in the system parameters caused due to externalexcitation The system parameters can be inertia or stiffnessThe response of the system is orthogonal to the direction ofexternal excitation System undergoes parametric resonancewhen the external excitation is equal to integral multiple

of natural frequency of the system In parametric reso-nance systems amplitude increases exponentially and maygrow without limit This exponential unlimited increase ofamplitude is potentially dangerous to the system Parametricresonance is also known as parametric instability or dynamicinstability Although parametric instability is secondary thesystem may undergo failure near the critical frequencies ofparametric instability Parametric instability is of concern instructures aircrafts and marine crafts Structural systemswhen subjected to vertical ground motion may undergoparametric resonance leading to largemotion or deformationof structure due to which structure may damage or collapseThe problem of parametric instability in various structuresis studied and well documented by Bolotin [1] Parametricresonance also occurs in case of liquid sloshing which is thepresent interest of the paper

Themotion of the unrestrained free-surface of the liquiddue to external excitation in the liquid-filled containers isknown as sloshing Sloshing is likely to be seen whenever wehave a liquid with a free-surface in the presence of gravityAt equilibrium the free surface of the liquid is static whenthe container is perturbed an oscillation is set up in the

2 Modelling and Simulation in Engineering

free surface The phenomenon of liquid sloshing occurs ina variety of engineering applications such as sloshing inliquid-propellant launch vehicles liquid oscillation in largestorage tanks by earthquake sloshing in the nuclear reactorsof pool type nuclear fuel storage tanks under earthquakeand water flow on the deck of the ship Such liquid motionis potentially dangerous problem to engineering structuresand environment leading to failure of engineering structuresand unexpected instabilityThus understanding the dynamicbehavior of liquid free surface is essential As a result theproblem of sloshing has attracted many researchers andengineersmotivating to understand the complex behaviour ofsloshing and to design the structures to withstand its effects

Liquid sloshing can be stimulated by a variety of containerexcitations The container excitation can be horizontal verti-cal or rotational Under horizontal excitations the liquid freesurface experiences normal sloshing the sloshing frequencywill be equal to excitation frequencyWhen the external exci-tation frequency is equal to fundamental sloshing frequencythe free surface undergoes resonance Extensive researchhas been done on sloshing response under pure horizontalexcitations When the liquid-filled container is subjected tovertical excitations for some combinations of amplitude andfrequency of the external excitation the free surface under-goes unbounded motion leading to parametric instabilityand for few other combinations the free surface undergoesbounded motion In the present paper the response of thefluid-filled containers under vertical excitation is undertaken

The problem of liquid response under vertical excitationswas first studied experimentally by Faraday [2] reporting thatthe frequency of the liquid vibrations on free surface is halfof the external excitation frequency The sloshing waves gen-erated under vertical excitation are sometimes referred as toFaraday waves Matthiessen [3] conducted experiments andreported that the fluid free surface vibrations are synchronousto the external excitation The Faraday study has been ana-lyzed by Rayleigh [4 5] and he confirmed Faradayrsquos obser-vationsThe discrepancy between Faradayrsquos observations andMatthiessenrsquos observations was explained mathematically byBenjamin andUrsell [6] Benjamin andUrsell [6] investigatedthe problem theoreticallyThey considered linearized inviscidpotential flow model with surface tension They concludedthat the response of the plane free surface of fluid undervertical excitation is governed by the Mathieu equationThe solution for the Mathieu equation [7] may be stableperiodic or unstable depending on the system parametersThe stability and instability of the Mathieu equation areshown in the form of plots of amplitude versus frequencyof external excitation which gives regions of stability andinstability Benjamin and Ursell [6] concluded that if theresponse of the free-surface is unstable the resulting motioncan have a frequency equal to (12 )119899Ω where 119899 is aninteger and Ω is the natural sloshing frequency Dodge et al[8] analyzed the sloshing response in propellant tanks ofspace vehicles theoretically and experimentally under verticalexcitations Mitchell [9] extended the studies on stabilityof free-surface of liquid under vertical random excitationsKhandelwal and Nigam [10] have studied the parametricinstability in rectangular tanks analytically Miles [11] studied

the parametric resonance considering nonlinear terms incylindrical tanks theoretically The problem of parametricoscillations was discussed by Miles and Henderson [12]in their review paper The study of Faraday waves is alsoreported in works by Perlin and Schultz [13] and Jiang etal [14] Most of the studies available in the old literature onthe problem of parametric sloshing under vertical excitationare experimental or theoretical involving complex equationsand heavy derivations In order to understand the complexbehaviour of sloshing under vertical excitations includingnonlinear terms numerical simulations are advantageouscompared to analytical solutions Analytical solutions getcomplicated when the shape of container is not regularNumerical methods like finite element method finite vol-ume method finite difference method boundary elementmethod and so forth are available for numerical modeling ofsloshing waves Wu et al [15] applied finite element methodfor solving sloshing problem Wu et al also discussed thesloshing response under vertical excitations in rectangulartanks using a 3D model Frandsen [16] analyzed the problemnumerically and theoretically considering fully nonlinearinviscid potential flow equations Frandsen applied modifiedsigma-transformed finite difference method for sloshingresponse Frandsen study was on 2-dimensional rectangulartanks Ning et al [17] applied boundary element methodto study the liquid sloshing in rectangular containers undercoupled horizontal and vertical excitation In the presentpaper the sloshing response under vertical excitations inrectangular tank is taken up First the stability of plane freesurface of liquid in rectangular tank is obtained from thelinearized equations and then the obtained stability chartis validated numerically applying finite element method Theobjective of the present paper is to study the stability of free-surface of liquid in rectangular tank under vertical excitationsnumericallyThe tank is assumed to be rigid with aspect ratio(ℎ119871) of 05 ℎ is depth of fluid and 119871 is length of the tankFirst the stability of the free-surface of the liquid is analyzedtheoretically through the governing linearized equationsStability chart for the liquid free-surface is plotted fromthese linearized equations using harmonic balance methodThe plotted stability chart is validated numerically A finiteelement numerical formulation based on nonlinear potentialtheory is developed for simulating sloshing response

2 Governing Equations

Consider a rectangular tank fixed in the Cartesian coordinatesystem 119874119909119911 which is moving with respect to the inertialCartesian coordinate system 119874

011990901199110vertically along the 119911-

direction The origins of these coordinates systems are at theleft end of the tank wall at the free surface and pointingupwards to the 119911-direction These two Cartesian coordinatesystems coincide when the tank is at rest Figure 1 shows thetank in the moving Cartesian coordinate system 119874119909119911 alongwith the prescribed boundary conditions

Let the displacement of the tank be governed by thedirections of axes as

119911 = 119911119905(119905) (1)

Modelling and Simulation in Engineering 3

X

Z

n

nL

ΩΔ2120601 = 0

120577(x t)Γs 120601(x z t)

h n

zt(t) = acos(120596t)

ΓB 120597120601120597n

= 0

nabla

Figure 1 Sloshing tank and its boundary condition

Fluid is assumed to be inviscid incompressible and irrota-tional Therefore the fluid motion is governed by Laplacersquosequation with the unknown velocity potentialΦ as follows

nabla2120601 = 0 (2)

Fluid obeys the Neumann boundary conditions on the wallsof the container and the Dirichlet boundary condition atthe liquid free surface In the moving coordinate system thevelocity component of the fluid normal to the walls is zeroHence at the bottom and on the walls of the tank (Γ

119861) we

have that120597120601

120597119899

10038161003816100381610038161003816100381610038161003816119909=0119871= 0

120597120601

120597119899

10038161003816100381610038161003816100381610038161003816119911=minusℎ= 0 (3)

On the free surface (Γ119878) dynamic and kinematic boundary

conditions hold and they are given by

120597120601

120597119905

10038161003816100381610038161003816100381610038161003816119911=120577+1

2nablaΦ sdot nablaΦ + (119892 + 119911

10158401015840

119905) 120577 = 0 (4)

120597120577

120597119905+120597120601

120597119909

120597120577

120597119909minus120597120601

120597119911= 0 (5)

where 120577 is the free surface elevation measured verticallyabove still water level 11991110158401015840

119905is the vertical acceleration of the

tank and 119892 is the acceleration due to gravityEquations (1)ndash(5) give the complete behaviour of non-

linear sloshing in fluids under vertical base excitation of thetank The position of the fluid free surface is not known apriori to solve the problem the fluid is assumed to be at restwith some initial perturbation on the free surface Thus theinitial conditions for the free surface in the moving Cartesiansystem at 119905 = 0 and 119911 = 0 can be written as

Φ (119909 0 0) = minus119911119889119911119905(119905)

119889119905 (6)

120577 (119909 0) = 1205770 (7)

where 1205770is the initial elevation of the free surface It should

be noted that it is not possible to attain the initial boundarycondition (7) maintaining (6) in reality it is a nonphysicalconditionThis condition is used in case of vertical excitationsalone because in this excitation some initial perturbation isneeded without which there will not be any oscillation in thefluid free surface

3 Governing Equation for DynamicStability of Free-Surface

In this section the governing equation for dynamic stabilityof free-surface of liquid under vertical excitation is derivedThe general solution for the Laplace equation in the rectan-gular domain satisfying the rigid boundary conditions (5) canbe written as

120601 =

infin

sum119899=0

cosh (119896119899(119911 + ℎ))

cosh (119896119899ℎ)

cos (119896119899119909) 119865119899 (119905)

120577 =

infin

sum119899=0

cos (119896119899119909) 119911119899(119905)

(8)

where 119896119899= 119899120587119871 is the wave number for mode number

119899 119865119899(119905) 119885

119899(119905) are the time evolution functions of the

respective 119899th mode and can be calculated by substitutingthe general solution (8) in the linear free-surface boundaryconditions obtained from (4) and (5) The linearized free-surface boundary conditions are

120597120601

120597119905

10038161003816100381610038161003816100381610038161003816119911=120577+ (119892 + 119911

10158401015840

119905) 120577 = 0

120597120577

120597119905minus120597120601

120597119911= 0

(9)

Substituting (8) into (9) leads to

119889119865119899(119905)

119889119905+ (119892 + 119911

10158401015840

119905) 119911119899(119905) = 0 (10)

119889119911119899(119905)

119889119905minus 119896119899tanh (119896

119899ℎ) 119865119899(119905) = 0 (11)

Substituting (10) into (11) and arranging terms

1198892119911119899(119905)

1198891199052+ 1205962

119899(1 +

11991110158401015840119905

119892)119911119899(119905) = 0 (12)

where

120596119899= radic119892119896

119899tanh (119896

119899ℎ) (13)

Equation (13) gives the linear sloshing frequenciesThe tank isassumed to be subjected to harmonic vertical excitation givenas

119911119905(119905) = 119886V cos (120596V119905) (14)

Equation (11) reduces to

1198892119911119899(119879)

1198891198792+ Ω2

119899(1 minus 119896V cos (2119879)) 119911119899 (119879) = 0 (15)

where 119879 = 12120596V119905 Ω119899 = 120596119899120596V and 119896V = 119886V1205962

V119892 Equation(15) is a Mathieu equation The stability and instability of thefree-surface are guided by (15)

TheMathieu equation is a second-order differential equa-tion with periodic coefficients The solution for the Mathieu


equation may be bounded or unbounded that is stable orunstable If the solution is bounded it may be periodic ornonperiodic The theory of the Mathieu equation is welldocumented by Mc Lachlan [7] the only reference in theliterature which covers exclusively theMathieu equationTheformof the solution for theMathieu equation can be obtainedby Floquetrsquos theory [18] According to the Floquet theory thesolution for theMathieu equation can be expressed as a linearcombination of two linearly independent Floquet solutions119865119903(119911) and 119865

119903(minus119911) The Floquet solution can be represented in

the form

119865119903(119911) = 119890

119894119903119911119901 (119911) (16)

where 119903 is called the characteristic exponent and depends onΩ119899 119896V and 119901(119911) is a periodic function with period 120587 The

behaviour of the solution can be obtained from (16) Solutionfor the Mathieu equation is bounded when the value of 119903 isreal and unbounded when the value of 119903 is complex In stablesolution if 119903 is irrational then solution is not periodic if 119903is rational then solution is periodic but not with period 120587 or2120587 and if 119903 is an integer then solution is periodic with period120587 or 2120587

The behaviour of the solution can be obtained from thestability chart a plot of system parameters Ω

119899 119896V The plots

have regions of stability and instability fromwhich behaviourof solution can be sought The stable and unstable regionsare separated by boundary curves on which the solution isperiodic with period 120587 or 2120587 To plot the stability chart itis needed to plot only the boundary curves The periodicsolutions on the boundary curves can be expanded as aFourier series [1] The periodic solution with a period 2120587 canbe written in the form

119911 (119905) =

infin

sum119896=135

(119886119896sin 119896119905

2+ 119887119896cos 119896119905

2) (17)

Substituting the series (17) into (15) and equating the coeffi-cients of identical sine and cosine terms lead to the followingsystem of linear homogenous algebraic equations

(1 +119896V

2minus1

4Ω2119899

)1198861minus119896V

21198863= 0 (18)

(1 minus1198962

4Ω2119899

)119886119896minus119896V

2(119886119896minus2+ 119886119896+2) = 0 119896 = 3 5 7

(19)

(1 minus119896V

2minus1

4Ω2119899

)1198871minus119896V

21198873= 0 (20)

(1 minus1198962

4Ω2119899

)119887119896minus119896V

2(119887119896minus2+ 119887119896+2) = 0 119896 = 3 5 7

(21)

The periodic solution with period 120587 can be expressed in theFourier series [1] as

119911 (119905) = 1198870 +

infin

sum119896=246

(119886119896sin 119896119905

2+ 119887119896cos 119896119905

2) (22)


(1 minus1

Ω2119899

)1198862minus119896V

21198864= 0 (23)

(1 minus1198962

4Ω2119899

)119886119896minus119896V

2(119886119896minus2+ 119886119896+2) = 0 119896 = 4 6 8

(24)

1198870minus119896V

21198872= 0

(1 minus1

Ω2119899

)1198872minus119896V

2(1198870+ 1198874) = 0

(25)

(1 minus1198962

4Ω2119899

)119887119896minus119896V

2(119887119896minus2+ 119887119896+2) = 0 119896 = 4 6 8

(26)

The system of linear homogenous algebraic equation (19)and (21) and (24) and (26) has a nontrivial solution whenthe determinant composed of the coefficients is zero Thedeterminants are written as

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 plusmn119896V

2minus1

4Ω2119899

minus119896V

20 sdot sdot sdot

minus119896V

21 minus

9

4Ω2119899

minus119896V

2sdot sdot sdot

0 minus119896V

21 minus

25

4Ω2119899

sdot sdot sdot

d

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (27)

Equation (27) gives the determinant obtained fromboth (19)ndash(21) combined under the plusmn sign as follows

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 minus1

Ω2119899

minus119896V

20 sdot sdot sdot

minus119896V

21 minus

4

Ω2119899

minus119896V

2sdot sdot sdot

0 minus119896V

21 minus

16

Ω2119899

sdot sdot sdot

d

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 minus119896V

20 0 sdot sdot sdot

minus119896V 1 minus1

Ω2119899

minus119896V

20 sdot sdot sdot

0 minus119896V

21 minus

4

Ω2119899

minus119896V

2sdot sdot sdot

0 0 minus119896V

21 minus

16

Ω2119899

sdot sdot sdot

d

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0

(28)

The determinants given above are called the Hill deter-minants and they are of infinite orderThe Hill determinants


are tridiagonal in nature It is clear from these determinantsthat Ω

119899always appear on the diagonal of the matrices one

can invoke the analogy with the eigenvalue problem andrefer to Ω

119899as an eigenvalue Then for given values of 119896V

it is possible to calculate values of Ω119899corresponding to

periodic solutions of periods 120587 and 2120587 By solving the aboveeigenvalue problems a stability chart is plotted The stabilitychart for (15) is shown in Figure 2 From Figure 2 we canpredict the stability of the free-surface If the amplitude andfrequency of the external excitation lie inside the curves(grey-colored region) the free-surface is unstable and theamplitude grows unbounded exponentially If the parametersof external excitation are outside the curves (white region)the free-surface is stable Thus from the stability chart thestability of the free-surface can be predicted

4 Numerical Procedure

In the present section the response of the free-surface undervertical excitations of the tank is simulated numericallyusing finite element method The finite element model basedon the mixed Eulerian-Lagrangian scheme is adopted Thefree surface nodes behave like the Lagrangian particles andthe interior nodes behave like the Eulerian particles Forthis formulation the free-surface kinematic and boundaryconditions (4) and (5) respectively are modified and writtenin the Lagrangian form as [19]

119889Φ

119889119905

10038161003816100381610038161003816100381610038161003816119911=120577=1

2nablaΦ sdot nablaΦ minus (119892 + 119911

10158401015840

119905) 120577 (29)

119889119909

119889119905=120597Φ

120597119909

119889119911

119889119905=120597Φ

120597119911 (30)

The problem of sloshing is nonlinear because the free-surface position is not known a priori and the boundaryconditions have nonlinear terms The sloshing problem isevaluated as an initial boundary value problem The initialboundary is taken as

Φ (119909 0 0) = minus119911119889119911119905(119905)

119889119905= 0

120577 (119909 0) = 1205770= 119886 cos (119896

119899119909)

(31)

where 119886 is the amplitude of the initial perturbation In orderto solve this nonlinear sloshing problem a finite elementnumerical approach based on themixed Eulerian-Lagrangianscheme is employed The free surface nodes behave likethe Lagrangian particles and interior nodes behave like theEulerian particles A four-node isoparametric element isused in the analysis Time interval 119905 is divided into finitenumber of time steps 119905

119899= 119899Δ119905 (119899 = 0 1 2 3 ) at a

particular time step (119899 = 0) the initial boundary conditions(31) are known using these initial conditions along withthe boundary condition (3) the Laplace equation (2) issolved to get velocity potential Φ with which velocity Vis evaluated with these evaluated velocities the kinematicand dynamic free surface boundary conditions (29)-(30) aretime integrated and the position of free surface is updated to

0 02 04 06 08 1 12 140

02

04

06

08

12

14

16

18

2

1

Unstable regionStable region

k = a1205962g

Ωn=120596n120596

Figure 2 Stability chart for dynamic stability of free-surface undervertical excitations

get the free surface position for the next time step (119899 = 1) Inthis manner the sloshing response is numerically simulatedThe velocities from the potential are calculated using patchrecovery [20] The fourth-order Runge-Kutta method isemployed to advance the solution in time As the timeproceeds in the simulation due to the Lagrangian behaviourof the free surface nodes these nodes move closer anddevelop a steep gradient leading to numerical instability Toget rid of this problem a cubic spline interpolation is used toregrid the free-surface uniformlyThe authors have developeda finite element numerical formulation for nonlinear sloshingresponse for sloshing in 2D rectangular tanks and verticalaxisymmetric tanks under horizontal excitations and verticalcoupled excitations [21 22]The same numerical formulationis followed here the present simulation is an extension of[21 22] sloshing response under vertical excitations In thepresent paper the numerical formulation is explained brieflyfor complete in depth explanation and algorithm readers canrefer to [21 22]

41 Finite Element Formulation The solution for the non-linear sloshing boundary value problem is obtained usingfinite elementmethodThe entire liquid domain is discretizedby using four-node isoparametric quadrilateral elements Atypical mesh structure assuming some initial perturbation onthe free-surface is shown in Figure 3 By introducing the finiteelement shape functions the liquid velocity potential can beapproximated as

120601 (119909 119911) =

119899

sum119895=1

119873119895(119909 119911) 120601

119895 (32)

where 119873119895is the shape function 119899 is the number of nodes in

the element and 120601119895is the nodal velocity potential


Figure 3 Finite element discretization of liquid domain

The potential on the free surface at a particular time stepis obtained from the free surface boundary condition (like at119905 = 0 (6)) It is needed to calculate the velocity potential forthe interior nodes Applying the Galerkin residual methodto the Laplace equation along with the Neumann boundaryconditions and taking the free surface nodes where thepotential is known to the right-hand side will lead to thefollowing system of finite element equations

intΩ

nabla119873119894

119898

sum119895=1

Φ119895nabla119873119895119889Ω|119895119894notinΓ

119904

= minusintΩ

nabla119873119894

119898

sum119895=1

120601119895nabla119873119895119889Ω|119895isinΓ

119904119894notinΓ119904

(33)

where119898 is the total number of nodes in the liquid domain

42 Velocity Recovery To track the free surface (29) and(30) need velocities which are computed from the calculatedpotential field using

V = nabla119873119895sdot Φ119895 (34)

The velocities calculated using (34) are the velocities at theGauss integration points and they do not possess interele-ment continuity and have a low accuracy at nodes andelement boundaries Utmost care should be taken to calculatethe velocities a small error in the velocity recovery willaffect the accuracy of free surface updating or tracking getaccumulated with time and lead to underestimation of thesolution In order to derive a smoothed and continuousvelocity patch recovery technique [20] is applied In patchrecovery technique the continuous velocity field is obtainedby considering the linear interpolation of the velocities at theGauss integration points as follows

V = 1198861+ 1198862120585 + 1198863120578 + 1198864120585120578 (35)

where V is any velocity component (V119909or V119910) 120585 120578 are the

Gauss locations and 1198861 1198862 1198863 and 119886

4are unknowns which

need to be evaluated To evaluate these unknowns a leastsquare fit is considered between V and V as follows

119865 (119886) =

4

sum119894=1

V (120585119894 120578119894) minus V (120585

119894 120578119894)2 (36)

where 119894 is the 2 times 2-order Gauss integration points Thenthe four unknown coefficients are determined from foursimultaneous equations obtained from

120597119865 (119886)

120597119886119896

= 0 119896 = 1 2 3 4 (37)

Substituting these calculated 119886119896rsquos in (35) gives the velocity

values for individual elements and these are averaged for thecommon nodes Finally a smoothed velocity field which isinterelement continuous is constructed by interpolating thefinite element shape functions used in (32) and the nodalaveraged velocities The global continuous velocity field isgiven as

V = 119873 sdot V (38)

where V is a velocity component (V119909or V119910)

43 Numerical Time Integration and Free Surface TrackingAfter calculating the velocity at a time step 119905 it is required tocalculate the position of free surface from (30) and determinethe potential on the free surface using (29) for the next timestep 119905 + Δ119905 As a result the liquid mesh and the boundarycondition required for the next time step are establishedThis is done using a finite difference numerical procedureThe numerical time integration scheme plays a major role inany time marching problem The fourth-order Runge-Kuttaexplicit time integration method is employed in the presentpaper The nodal coordinates of the free surface and theassociated velocity potential at a current time step 119894 are knownand can be represented in a single variable as

119904119894= (119909119894 119911119894 Φ119894) (39)

where119909119894= (1199091 1199092 119909

119873119883+1)119894

119911119894= (1199111 1199112 119911

119873119883+1)119894

Φ119894= (Φ1 Φ2 Φ

119873119883+1)119894

(40)

where 119873119883 is number of segments along the free surfaceSimilarly the time derivative can be written as

119863119904119894

119863119905= 119865 (119905

119894 119904119894) = 119865119894 (41)

The free surface position and the associated velocity potentialat the next time step 119894 + 1 can be expressed as

119904119894+1= 119904119894+1199041

6+1199042

3+1199043

3+1199044

6 (42)

where1199041= nabla119905119865 (119905

119894 119904119894)

1199042= nabla119905119865 (119905

119894+nabla119905

2 119904119894+1199041

2)

1199043= nabla119905119865 (119905

119894+nabla119905

2 119904119894+1199042

2)

1199044= nabla119905119865 (119905

119894+ nabla119905 119904

119894+ 1199043)

(43)


After obtaining the new positions and potential of the freesurface the liquid domain is remeshed based on theseobtained new coordinate positions

44 Regridding Algorithm At the beginning of the numericalsimulation the free surface nodes are uniformly distributedalong the 119909-direction As the time proceeds the free surfacenodes are spaced unequally and they cluster into a steep gra-dient leading to numerical instabilityThis problemoccurs fora long-time simulation to avoid this instability an automaticregridding condition using cubic B-spline interpolation isemployed when the movement of the nodes is 75 more orless than the initial grid spacing For the regridding first thefree surface length 119871

119891is obtained Then the free surface is

divided into119873119883 segments with the identical arc length Thecoordinates of node are denoted as (119909

119894 119910119894) (119894 = 1 2 119873119883+

1) and let the arc length between two successive points 119894 and119894 + 1 be 119878

119894 Being a uniform regridding 119878

119894can be expressed as

119878119894=119894119871119891

119873119883 (44)

The coordinates of the nodes (119909119894 119910119894) are a function of the arc

length 119878119894

(119909119894 119911119894) = 119891 (119878

119894) (45)

The cubic B-spline interpolation is used to calculate thecoordinates (119909

119894 119911119894) and the velocity potential on the new

uniform free surface is also obtained in a similar fashion

45 Complete Algorithm for Sloshing Response Including allof the steps above the algorithm for numerical procedure ofnonlinear sloshing is as follows

(1) Assume the initial free surface elevation and theassociated velocity potential from (31)

(2) The liquid domain is discretized using finite elementsand the nodal connectivity data and nodal positionsdata are obtained

(3) Velocity potential for the interior nodes is calculatedusing (33)

(4) Recover the velocity from the velocity potential usingpatch recovery technique

(5) Update the free surface nodes using (29) and (30)to a new position and find the new associatedvelocity potential using the fourth-order Runge-Kuttamethod

(6) Check for the requirement of regridding(7) After finding the new free surface position and the

associated velocity potential repeat step 2 until thefinal time step is reached

5 Numerical Results and Discussion

A code is developed following the pervious numerical for-mulation for simulating the dynamic stability of free-surface

Table 1 Excitation parameters for the test cases shown in Figure 4

Case Ω119899

119896V 120596V 119886V1205962

V

1 1253 05 079811205961

05 g2 05 03 2120596

103 g

3 10 05 1205961

05 g4 05 02 2120596

302 g

5 06 05 1661205961

05 g6 055 05 18182120596

105 g

1

2

3

4

56

0 02 04 06 08 1 12 14k = a120596

2g

Ωn=120596n120596

02

04

06

08

12

14

16

18

2

1

Figure 4 Stability chart of sloshing response under vertical excita-tions (test cases are also shown)

sloshing response under various external vertical excitationamplitudes and frequencies The liquid sloshing responseinside a 20m wide rectangular rigid container with liquidfilled to a depth of 1m is simulated Aspect ratio of 05is maintained For the present tank the order of sloshingfrequencies from (13) is 120596

1= 37594 rads and 120596

3=

67986 rads The tank is assumed to be subjected to thevertical harmonic excitation given by (14) In the numericalsimulations 40 nodes along the 119909-direction and 20 nodesalong the 119911-direction are taken and a time step of 001 sis adopted The sloshing response is simulated for differentcases with increasing wave steepness inside and outside theregions of parametric instability or parametric resonanceThe steepness parameter depends on the initial conditionsconsidered 120576 = 119886 minus 1205962

119899119892 [16] The different cases considered

in simulation are marked on the stability chart as shown inFigure 4 The excitation parameters for the cases shown inFigure 4 are given in Table 1 The test cases considered aresimilar to the cases considered by Frandsen [16]

The first test case is in stable region with frequency ratioΩ1= 1253 and forcing amplitude 119896V = 05 test case 1 as

shown in Figure 4 The sloshing responses on the left wallof the tank for low (120576 = 00014) and high (120576 = 0288) wavesteepness are shown in Figures 5(a) and 5(b) respectivelyFigures 5(c) and 5(d) show the respective phase-plane plotsfor the small and steep wave cases The time histories of the


0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(a)

0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(b)

minus2 minus15 minus1 minus05 0 05 1 15 2minus2

minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(c)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(d)

Figure 5 Free-surface elevation on the left wall in the stable region (test case 1) Ω1= 1253 119896V = 05 for (a) 120576 = 00014 (b) 120576 = 0288 and

the respective phase-plane plots

free-surface elevation are nondimensionalized The sloshingresponse obtained with the present simulation is comparedwith numerical results of Frandsen [16] Both results are inexcellent agreement The sloshing response for low steepwaves is symmetric that is amplitudes of peaks and troughsare equal whereas for high steep waves the response isasymmetric showing different amplitudes for peaks andtroughs This is an indication of nonlinear response Thisnonlinear behaviour can be noticed from the respectivephase-plane plots The phase-plane plot of the sloshingresponse with low wave steepness shown in Figure 5(c) hasa closed orbit displaying a linear behaviour whereas thephase-plane plot of the sloshing response with high wavesteepness has nonrepeatable and nonclosed orbits displayinga nonlinear characteristic

The second test case lies in the unstable region withfrequency ratioΩ

1= 05 and forcing amplitude 119896V = 03 test

case 2 as shown in Figure 4 Figure 6(a) shows the free surfaceelevation on the left wall of the tank for a low wave steepnessof 120576 = 00014 As the excitation parameters lie in unstableregion an unbounded response is expected the sloshingresponse plot displays the expected behaviour Figure 6(b)shows the corresponding phase-plane diagram The phase-plane plot and the response plot clearly show that thenonlinear effects are predominant Amovingmesh generatedat different time steps in this unstable region for parametricresonance is shown in Figure 7 The animation of simulationin this case can be seen in supplementary file Dynamic Insta-bilitygif see the supplementary file in Supplementary Mate-rial available online at httpdxdoiorg1011552013252760


0 10 20 30 40 50 60 70 80 90minus400

minus200

0

200

400

600

800

t1205961

120577(0h)a

(a)

minus400 minus200 0 200 400 600 800

minus800

minus600

minus400

minus200

0

200

400

600

120577a

120597120577120597ta1205961

(b)

Figure 6 Free-surface elevation on the left wall in the unstable region (test case 2) for Ω1= 05 119896V = 05 120576 = 00014 and the respective

phase-plane plot

t = 5 s

(a)

t = 10 s

(b)

t = 15 s

(c)

t = 18 s

(d)

t = 20 s

(e)

t = 21 s

(f)

t = 22 s

(g)

t = 23 s

(h)

t = 24 s

(i)

Figure 7 Moving mesh generated at different type steps in parametric resonance


0 50 100 150 200 250 300 350 400minus150

minus100

minus50

0

50

100

150

200

250

t1205961

120577(0h)a

(a)

0 50 100 150 200 250 300minus100

minus50

0

50

100

150

120577(0h)a

t1205961

(b)

minus150 minus100 minus50 0 50 100 150 200minus250

minus200

minus150

minus100

minus50

0

50

100

150

120577a

120597120577120597ta1205961

(c)

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

minus100 minus50 0 50 100 150120577a

120597120577120597ta1205961

(d)

Figure 8 Free-surface elevation on the left wall in the unstable regions with 120576 = 00014 for (a) Ω1= 10 119896V = 05 (test case 3) (b) Ω3 = 05

119896V = 04 (test case 4) and the respective phase-plane plots

Figure 8 also shows the sloshing response time historiesin unstable regions A low wave steepness parameter of 120576 =00014 is considered Figure 8(a) shows the sloshing responsetime history for frequency ratio Ω

1= 10 and forcing

amplitude 119896V = 05 test case 3 as shown in Figure 4 Thiscase corresponds to instability in the first sloshingmode lyingin the second instability region According to the theorythe effect of parametric resonance gradually reduces as wemove to the higher regions of instability As expected theamplitudes do not grow rapidly in this instability regioncompared to the first instability region response shown inFigure 6(a) First the amplitude of the sloshing responsestarts growing exponentially in a resonance mode and thenafter certain time the response reduces gradually As theamplitude increases the natural frequency of the systemchanges and creates low frequency amplitude oscillationsleading to decrease in amplitudes of responseThis behaviour

is called detuning effect under parametric excitation offrequency close to twice the natural frequency of a certainmode the free-surface oscillates exhibiting the shape of thatmode As the excitation amplitude increases the naturalfrequency changes and the input energy can excite the otherneighbor nodes If the excited neighbor nodes are stablethe increase in the amplitude will be suppressed leading todetuning effect This detuning effect can be captured only innonlinear systems In case of linear systems [6] the responsewill be always increasing this detuning effect cannot becaptured The present finite element nonlinear numericalmodel can capture this detuning effect effectively Figure 8(c)shows the respective phase-plane plot Figure 8(b) shows thesloshing response on left wall of tank for frequency ratioΩ3= 05 and forcing amplitude 119896V = 02 test case 4 as

shown in Figure 4 This case corresponds to instability inthe second sloshing mode lying in the first instability region


t = 40 s

(a)

t = 65 s

(b)

t = 70 s

(c)

t = 75 s

(d)

t = 80 s

(e)

t = 82 s

(f)

t = 85 s

(g)

t = 88 s

(h)

t = 90 s

(i)

Figure 9 Moving mesh generated at different time steps in unstable region displaying detuning effect

As the instability is in the secondmode the amplitudes do notgrow rapidly when compared to instability in the first modeFigure 6(a) After certain time the amplitude comes downshowing the detuning effect In this case the free-surfaceoscillates exhibiting the third sloshing mode and as theamplitude increases the input parametric excitation excitesthe first sloshing mode which is stable and the amplitudesfall down Similar free-surface behaviour is reported byFrandsen [16] Figure 8(d) shows the respective phase-planeplot The phase-plane plots for the responses display a linearbehaviour for the system Figure 9 shows the moving meshgenerated for the response shown in Figure 8(b) at varioustime steps The animation of simulation in this case can beseen in supplementary file Detuning Effectgif

Figure 10 shows the sloshing response for test case 5 asshown in Figure 4 with frequency ratioΩ

1= 06 and forcing

amplitude 119896V = 05 This point lies in the stable regionbut very close to instability region Figure 10(a) shows thesloshing response for low wave steepness parameter 120576 =

00014 and Figure 10(b) shows the sloshing response forhigh wave steepness parameter 120576 = 0288 As expected thepoint is in the stable region and the sloshing response isstable

Figure 11 shows the sloshing response for test case 6 withfrequency ratio Ω

1= 055 and forcing amplitude 119896V = 05

lying in the unstable region as shown in Figure 4 A low

steepness parameter 120576 = 00014 is taken As the point lies inthe unstable region the response is also unstable as expectedThe low steepness response is sufficient to be demonstratedto show the rapid increase in the amplitudes

6 Conclusion

The free-surface of fluid under vertical excitation of the tankreduces to the Mathieu equation Free-surface undergoesparametric resonance for some combinations of verticalexcitation frequencies and amplitudes this resonance ischaracterized by exponentially unbounded growth of ampli-tude Stability chart is plotted for the dynamic stabilityanalysis from linear equations A fully nonlinear finite ele-ment numerical model has been developed based on thepotential flow theory for simulating the sloshing responseThe sloshing response is simulated for different cases lyingin the stability chart In the stable regions the free-surfaceresponse is always bounded In the unstable regions thefree-surface undergoes parametric resonance characterizedby an unbounded response of the free-surface In the unstableregions even small excitations can cause the growth of smallinitial perturbations if the tank is excited for a sufficientlylong time The sloshing response obtained is in exact agree-ment with the theoretical predictions of the stability chartThe present numerical model can capture detuning effects


0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(a)

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(b)

Figure 10 Free-surface elevation on the left wall in the stable region (test case 5) for Ω1= 055 119896V = 05 (a) 120576 = 00014 and (b) 120576 = 0288

0 10 20 30 40 50 60 70minus400

minus300

minus200

minus100

0

100

200

300

400

500

600

t1205961

120577(0h)a

Figure 11 Free-surface elevation on the left wall in unstable region(test case 6) forΩ

1= 055 119896V = 05 and 120576 = 00014

as well Detuning is due to change of frequencies during theamplitude growth

References

[1] V V Bolotin The Dynamic Stability of Elastic Systems HoldenDay San Francisco Calif USA 1964

[2] M Faraday ldquoOn a peculiar class of acoustical figures and oncertain forms assumed by groups of particles upon vibratingelastic surfacesrdquo Philosophical Transactions of the Royal SocietyB vol 121 pp 299ndash340 1831

[3] L Matthiessen ldquoAkustische Versuche die kleinsten Transversalwellen der Flussigskeitcn bctressendrdquoAnnals of Physics vol 134pp 107ndash117 1868

[4] L Rayleigh ldquoOn maintained vibrationsrdquo Philosophical Maga-zine vol 15 pp 229ndash235 1883

[5] L Rayleigh ldquoOn the crispations of fluid resting upon a vibratingsupportrdquo Philosophical Magazine vol 16 pp 50ndash58 1883

[6] T B Benjamin and F Ursell ldquoThe stability of the plane freesurface of a liquid in vertical periodic motionrdquo Proceedings ofthe Royal Society of London A vol 225 pp 505ndash515 1954

[7] N W Mc Lachlan Theory and Applications of Mathieu Func-tions Oxford University Press New York NY USA 1957

[8] F T Dodge D D Kana and H N Abramson Liquid SurfaceOscillations in Longitudinally Excited Rigid Cylindrical Contain-ers NASA CR-56135 1964

[9] R R Mitchell Stochastic Stability of the Liquid Free Surface inVertically Excited Cylinders Analytical Dynamics LaboratoryAuburn Ala USA 1968

[10] R S Khandelwal andN C Nigam ldquoParametric instabilities of aliquid free surface in a flexible container under vertical periodicmotionrdquo Journal of Sound and Vibration vol 74 no 2 pp 243ndash249 1981

[11] J W Miles ldquoNonlinear Faraday resonancerdquo Journal of FluidMechanics vol 146 pp 285ndash302 1984

[12] J Miles and D Henderson ldquoParametrically forced surfacewavesrdquoAnnual Review of FluidMechanics vol 22 no 1 pp 143ndash165 1990

[13] M Perlin and W W Schultz ldquoCapillary effects on surfacewavesrdquo Annual Review of Fluid Mechanics vol 32 pp 241ndash2742000

[14] L Jiang C-L Ting M Perlin and W W Schultz ldquoModerateand steep Faradaywaves instabilitiesmodulation and temporalasymmetriesrdquo Journal of Fluid Mechanics vol 329 pp 275ndash3071996

[15] G X Wu Q W Ma and R Eatock Taylor ldquoNumericalsimulation of sloshing waves in a 3D tank based on a finiteelement methodrdquo Applied Ocean Research vol 20 no 6 pp337ndash355 1998

[16] J B Frandsen ldquoSloshing motions in excited tanksrdquo Journal ofComputational Physics vol 196 no 1 pp 53ndash87 2004


[17] D-Z Ning W-H Song Y-L Liu and B Teng ldquoA boundaryelement investigation of liquid sloshing in coupled horizontaland vertical excitationrdquo Journal of Applied Mathematics vol2012 Article ID 340640 20 pages 2012

[18] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Dover New York NY USA 1972

[19] M S Longuet-Higgins and E D Cokelet ldquoThe deformationof steep surface waves on water I A numerical method ofcomputationrdquo Proceedings of the Royal Society of London A vol350 pp 1ndash26 1976

[20] O C Zienkiewicz and J Z Zhu ldquoSuperconvergent patchrecovery and a posteriori error estimates Part 1 the recoverytechniquerdquo International Journal for Numerical Methods inEngineering vol 33 no 7 pp 1331ndash1364 1992

[21] S S Kolukula and P Chellapandi ldquoNonlinear finite elementanalysis of sloshingrdquo Advances in Numerical Analysis vol 2013Article ID 571528 10 pages 2013

[22] S S Kolukula and P Chellapandi ldquoDynamic stability of planefree surface of liquid in axisymmetric tanksrdquo Advances inAcoustics and Vibration vol 2013 Article ID 298458 16 pages2013

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free surface The phenomenon of liquid sloshing occurs ina variety of engineering applications such as sloshing inliquid-propellant launch vehicles liquid oscillation in largestorage tanks by earthquake sloshing in the nuclear reactorsof pool type nuclear fuel storage tanks under earthquakeand water flow on the deck of the ship Such liquid motionis potentially dangerous problem to engineering structuresand environment leading to failure of engineering structuresand unexpected instabilityThus understanding the dynamicbehavior of liquid free surface is essential As a result theproblem of sloshing has attracted many researchers andengineersmotivating to understand the complex behaviour ofsloshing and to design the structures to withstand its effects

Liquid sloshing can be stimulated by a variety of containerexcitations The container excitation can be horizontal verti-cal or rotational Under horizontal excitations the liquid freesurface experiences normal sloshing the sloshing frequencywill be equal to excitation frequencyWhen the external exci-tation frequency is equal to fundamental sloshing frequencythe free surface undergoes resonance Extensive researchhas been done on sloshing response under pure horizontalexcitations When the liquid-filled container is subjected tovertical excitations for some combinations of amplitude andfrequency of the external excitation the free surface under-goes unbounded motion leading to parametric instabilityand for few other combinations the free surface undergoesbounded motion In the present paper the response of thefluid-filled containers under vertical excitation is undertaken

The problem of liquid response under vertical excitationswas first studied experimentally by Faraday [2] reporting thatthe frequency of the liquid vibrations on free surface is halfof the external excitation frequency The sloshing waves gen-erated under vertical excitation are sometimes referred as toFaraday waves Matthiessen [3] conducted experiments andreported that the fluid free surface vibrations are synchronousto the external excitation The Faraday study has been ana-lyzed by Rayleigh [4 5] and he confirmed Faradayrsquos obser-vationsThe discrepancy between Faradayrsquos observations andMatthiessenrsquos observations was explained mathematically byBenjamin andUrsell [6] Benjamin andUrsell [6] investigatedthe problem theoreticallyThey considered linearized inviscidpotential flow model with surface tension They concludedthat the response of the plane free surface of fluid undervertical excitation is governed by the Mathieu equationThe solution for the Mathieu equation [7] may be stableperiodic or unstable depending on the system parametersThe stability and instability of the Mathieu equation areshown in the form of plots of amplitude versus frequencyof external excitation which gives regions of stability andinstability Benjamin and Ursell [6] concluded that if theresponse of the free-surface is unstable the resulting motioncan have a frequency equal to (12 )119899Ω where 119899 is aninteger and Ω is the natural sloshing frequency Dodge et al[8] analyzed the sloshing response in propellant tanks ofspace vehicles theoretically and experimentally under verticalexcitations Mitchell [9] extended the studies on stabilityof free-surface of liquid under vertical random excitationsKhandelwal and Nigam [10] have studied the parametricinstability in rectangular tanks analytically Miles [11] studied

the parametric resonance considering nonlinear terms incylindrical tanks theoretically The problem of parametricoscillations was discussed by Miles and Henderson [12]in their review paper The study of Faraday waves is alsoreported in works by Perlin and Schultz [13] and Jiang etal [14] Most of the studies available in the old literature onthe problem of parametric sloshing under vertical excitationare experimental or theoretical involving complex equationsand heavy derivations In order to understand the complexbehaviour of sloshing under vertical excitations includingnonlinear terms numerical simulations are advantageouscompared to analytical solutions Analytical solutions getcomplicated when the shape of container is not regularNumerical methods like finite element method finite vol-ume method finite difference method boundary elementmethod and so forth are available for numerical modeling ofsloshing waves Wu et al [15] applied finite element methodfor solving sloshing problem Wu et al also discussed thesloshing response under vertical excitations in rectangulartanks using a 3D model Frandsen [16] analyzed the problemnumerically and theoretically considering fully nonlinearinviscid potential flow equations Frandsen applied modifiedsigma-transformed finite difference method for sloshingresponse Frandsen study was on 2-dimensional rectangulartanks Ning et al [17] applied boundary element methodto study the liquid sloshing in rectangular containers undercoupled horizontal and vertical excitation In the presentpaper the sloshing response under vertical excitations inrectangular tank is taken up First the stability of plane freesurface of liquid in rectangular tank is obtained from thelinearized equations and then the obtained stability chartis validated numerically applying finite element method Theobjective of the present paper is to study the stability of free-surface of liquid in rectangular tank under vertical excitationsnumericallyThe tank is assumed to be rigid with aspect ratio(ℎ119871) of 05 ℎ is depth of fluid and 119871 is length of the tankFirst the stability of the free-surface of the liquid is analyzedtheoretically through the governing linearized equationsStability chart for the liquid free-surface is plotted fromthese linearized equations using harmonic balance methodThe plotted stability chart is validated numerically A finiteelement numerical formulation based on nonlinear potentialtheory is developed for simulating sloshing response

2 Governing Equations

Consider a rectangular tank fixed in the Cartesian coordinatesystem 119874119909119911 which is moving with respect to the inertialCartesian coordinate system 119874

011990901199110vertically along the 119911-

direction The origins of these coordinates systems are at theleft end of the tank wall at the free surface and pointingupwards to the 119911-direction These two Cartesian coordinatesystems coincide when the tank is at rest Figure 1 shows thetank in the moving Cartesian coordinate system 119874119909119911 alongwith the prescribed boundary conditions

Let the displacement of the tank be governed by thedirections of axes as

119911 = 119911119905(119905) (1)


X

Z

n

nL

ΩΔ2120601 = 0

120577(x t)Γs 120601(x z t)

h n

zt(t) = acos(120596t)

ΓB 120597120601120597n

= 0

nabla



nabla2120601 = 0 (2)


119861) we

have that120597120601

120597119899

10038161003816100381610038161003816100381610038161003816119909=0119871= 0

120597120601

120597119899

10038161003816100381610038161003816100381610038161003816119911=minusℎ= 0 (3)



120597120601

120597119905

10038161003816100381610038161003816100381610038161003816119911=120577+1


10158401015840

119905) 120577 = 0 (4)

120597120577

120597119905+120597120601

120597119909

120597120577

120597119909minus120597120601

120597119911= 0 (5)





Φ (119909 0 0) = minus119911119889119911119905(119905)

119889119905 (6)

120577 (119909 0) = 1205770 (7)





120601 =

infin

sum119899=0

cosh (119896119899(119911 + ℎ))

cosh (119896119899ℎ)

cos (119896119899119909) 119865119899 (119905)

120577 =

infin

sum119899=0

cos (119896119899119909) 119911119899(119905)

(8)


119899 119865119899(119905) 119885



120597120601

120597119905

10038161003816100381610038161003816100381610038161003816119911=120577+ (119892 + 119911

10158401015840

119905) 120577 = 0

120597120577

120597119905minus120597120601

120597119911= 0

(9)


119889119865119899(119905)

119889119905+ (119892 + 119911

10158401015840

119905) 119911119899(119905) = 0 (10)

119889119911119899(119905)

119889119905minus 119896119899tanh (119896

119899ℎ) 119865119899(119905) = 0 (11)


1198892119911119899(119905)

1198891199052+ 1205962

119899(1 +

11991110158401015840119905

119892)119911119899(119905) = 0 (12)

where

120596119899= radic119892119896

119899tanh (119896

119899ℎ) (13)


119911119905(119905) = 119886V cos (120596V119905) (14)


1198892119911119899(119879)

1198891198792+ Ω2

119899(1 minus 119896V cos (2119879)) 119911119899 (119879) = 0 (15)

where 119879 = 12120596V119905 Ω119899 = 120596119899120596V and 119896V = 119886V1205962






the form

119865119903(119911) = 119890

119894119903119911119901 (119911) (16)




119899 119896V The plots


119911 (119905) =

infin

sum119896=135

(119886119896sin 119896119905

2+ 119887119896cos 119896119905

2) (17)


(1 +119896V

2minus1

4Ω2119899

)1198861minus119896V

21198863= 0 (18)

(1 minus1198962

4Ω2119899

)119886119896minus119896V

2(119886119896minus2+ 119886119896+2) = 0 119896 = 3 5 7

(19)

(1 minus119896V

2minus1

4Ω2119899

)1198871minus119896V

21198873= 0 (20)

(1 minus1198962

4Ω2119899

)119887119896minus119896V

2(119887119896minus2+ 119887119896+2) = 0 119896 = 3 5 7

(21)


119911 (119905) = 1198870 +

infin

sum119896=246

(119886119896sin 119896119905

2+ 119887119896cos 119896119905

2) (22)


(1 minus1

Ω2119899

)1198862minus119896V

21198864= 0 (23)

(1 minus1198962

4Ω2119899

)119886119896minus119896V

2(119886119896minus2+ 119886119896+2) = 0 119896 = 4 6 8

(24)

1198870minus119896V

21198872= 0

(1 minus1

Ω2119899

)1198872minus119896V

2(1198870+ 1198874) = 0

(25)

(1 minus1198962

4Ω2119899

)119887119896minus119896V

2(119887119896minus2+ 119887119896+2) = 0 119896 = 4 6 8

(26)


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 plusmn119896V

2minus1

4Ω2119899

minus119896V

20 sdot sdot sdot

minus119896V

21 minus

9

4Ω2119899

minus119896V

2sdot sdot sdot

0 minus119896V

21 minus

25

4Ω2119899

sdot sdot sdot

d

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (27)


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 minus1

Ω2119899

minus119896V

20 sdot sdot sdot

minus119896V

21 minus

4

Ω2119899

minus119896V

2sdot sdot sdot

0 minus119896V

21 minus

16

Ω2119899

sdot sdot sdot

d

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 minus119896V

20 0 sdot sdot sdot


Ω2119899

minus119896V

20 sdot sdot sdot

0 minus119896V

21 minus

4

Ω2119899

minus119896V

2sdot sdot sdot

0 0 minus119896V

21 minus

16

Ω2119899

sdot sdot sdot

d

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0

(28)











119889Φ

119889119905

10038161003816100381610038161003816100381610038161003816119911=120577=1


10158401015840

119905) 120577 (29)

119889119909

119889119905=120597Φ

120597119909

119889119911

119889119905=120597Φ

120597119911 (30)


Φ (119909 0 0) = minus119911119889119911119905(119905)

119889119905= 0

120577 (119909 0) = 1205770= 119886 cos (119896

119899119909)

(31)


119899= 119899Δ119905 (119899 = 0 1 2 3 ) at a


0 02 04 06 08 1 12 140

02

04

06

08

12

14

16

18

2

1


k = a1205962g

Ωn=120596n120596




120601 (119909 119911) =

119899

sum119895=1

119873119895(119909 119911) 120601

119895 (32)






intΩ

nabla119873119894

119898

sum119895=1

Φ119895nabla119873119895119889Ω|119895119894notinΓ

119904

= minusintΩ

nabla119873119894

119898

sum119895=1

120601119895nabla119873119895119889Ω|119895isinΓ

119904119894notinΓ119904

(33)



V = nabla119873119895sdot Φ119895 (34)


V = 1198861+ 1198862120585 + 1198863120578 + 1198864120585120578 (35)



4are unknowns which


119865 (119886) =

4

sum119894=1

V (120585119894 120578119894) minus V (120585

119894 120578119894)2 (36)


120597119865 (119886)

120597119886119896

= 0 119896 = 1 2 3 4 (37)



V = 119873 sdot V (38)



119904119894= (119909119894 119911119894 Φ119894) (39)

where119909119894= (1199091 1199092 119909

119873119883+1)119894

119911119894= (1199111 1199112 119911

119873119883+1)119894

Φ119894= (Φ1 Φ2 Φ

119873119883+1)119894

(40)


119863119904119894

119863119905= 119865 (119905

119894 119904119894) = 119865119894 (41)


119904119894+1= 119904119894+1199041

6+1199042

3+1199043

3+1199044

6 (42)

where1199041= nabla119905119865 (119905

119894 119904119894)

1199042= nabla119905119865 (119905

119894+nabla119905

2 119904119894+1199041

2)

1199043= nabla119905119865 (119905

119894+nabla119905

2 119904119894+1199042

2)

1199044= nabla119905119865 (119905

119894+ nabla119905 119904

119894+ 1199043)

(43)






119894 119910119894) (119894 = 1 2 119873119883+




119878119894=119894119871119891

119873119883 (44)


length 119878119894

(119909119894 119911119894) = 119891 (119878

119894) (45)















Case Ω119899

119896V 120596V 119886V1205962

V

1 1253 05 079811205961

05 g2 05 03 2120596

103 g

3 10 05 1205961

05 g4 05 02 2120596

302 g

5 06 05 1661205961

05 g6 055 05 18182120596

105 g

1

2

3

4

56

0 02 04 06 08 1 12 14k = a120596

2g

Ωn=120596n120596

02

04

06

08

12

14

16

18

2

1



1= 37594 rads and 120596

3=







0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(a)

0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(b)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(c)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(d)








0 10 20 30 40 50 60 70 80 90minus400

minus200

0

200

400

600

800

t1205961

120577(0h)a

(a)

minus400 minus200 0 200 400 600 800

minus800

minus600

minus400

minus200

0

200

400

600

120577a

120597120577120597ta1205961

(b)


phase-plane plot

t = 5 s

(a)

t = 10 s

(b)

t = 15 s

(c)

t = 18 s

(d)

t = 20 s

(e)

t = 21 s

(f)

t = 22 s

(g)

t = 23 s

(h)

t = 24 s

(i)



0 50 100 150 200 250 300 350 400minus150

minus100

minus50

0

50

100

150

200

250

t1205961

120577(0h)a

(a)

0 50 100 150 200 250 300minus100

minus50

0

50

100

150

120577(0h)a

t1205961

(b)


minus200

minus150

minus100

minus50

0

50

100

150

120577a

120597120577120597ta1205961

(c)

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

minus100 minus50 0 50 100 150120577a

120597120577120597ta1205961

(d)




1= 10 and forcing





t = 40 s

(a)

t = 65 s

(b)

t = 70 s

(c)

t = 75 s

(d)

t = 80 s

(e)

t = 82 s

(f)

t = 85 s

(g)

t = 88 s

(h)

t = 90 s

(i)




1= 06 and forcing







6 Conclusion



0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(a)

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(b)


0 10 20 30 40 50 60 70minus400

minus300

minus200

minus100

0

100

200

300

400

500

600

t1205961

120577(0h)a


1= 055 119896V = 05 and 120576 = 00014


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










X

Z

n

nL

ΩΔ2120601 = 0

120577(x t)Γs 120601(x z t)

h n

zt(t) = acos(120596t)

ΓB 120597120601120597n

= 0

nabla



nabla2120601 = 0 (2)


119861) we

have that120597120601

120597119899

10038161003816100381610038161003816100381610038161003816119909=0119871= 0

120597120601

120597119899

10038161003816100381610038161003816100381610038161003816119911=minusℎ= 0 (3)



120597120601

120597119905

10038161003816100381610038161003816100381610038161003816119911=120577+1


10158401015840

119905) 120577 = 0 (4)

120597120577

120597119905+120597120601

120597119909

120597120577

120597119909minus120597120601

120597119911= 0 (5)





Φ (119909 0 0) = minus119911119889119911119905(119905)

119889119905 (6)

120577 (119909 0) = 1205770 (7)





120601 =

infin

sum119899=0

cosh (119896119899(119911 + ℎ))

cosh (119896119899ℎ)

cos (119896119899119909) 119865119899 (119905)

120577 =

infin

sum119899=0

cos (119896119899119909) 119911119899(119905)

(8)


119899 119865119899(119905) 119885



120597120601

120597119905

10038161003816100381610038161003816100381610038161003816119911=120577+ (119892 + 119911

10158401015840

119905) 120577 = 0

120597120577

120597119905minus120597120601

120597119911= 0

(9)


119889119865119899(119905)

119889119905+ (119892 + 119911

10158401015840

119905) 119911119899(119905) = 0 (10)

119889119911119899(119905)

119889119905minus 119896119899tanh (119896

119899ℎ) 119865119899(119905) = 0 (11)


1198892119911119899(119905)

1198891199052+ 1205962

119899(1 +

11991110158401015840119905

119892)119911119899(119905) = 0 (12)

where

120596119899= radic119892119896

119899tanh (119896

119899ℎ) (13)


119911119905(119905) = 119886V cos (120596V119905) (14)


1198892119911119899(119879)

1198891198792+ Ω2

119899(1 minus 119896V cos (2119879)) 119911119899 (119879) = 0 (15)

where 119879 = 12120596V119905 Ω119899 = 120596119899120596V and 119896V = 119886V1205962






the form

119865119903(119911) = 119890

119894119903119911119901 (119911) (16)




119899 119896V The plots


119911 (119905) =

infin

sum119896=135

(119886119896sin 119896119905

2+ 119887119896cos 119896119905

2) (17)


(1 +119896V

2minus1

4Ω2119899

)1198861minus119896V

21198863= 0 (18)

(1 minus1198962

4Ω2119899

)119886119896minus119896V

2(119886119896minus2+ 119886119896+2) = 0 119896 = 3 5 7

(19)

(1 minus119896V

2minus1

4Ω2119899

)1198871minus119896V

21198873= 0 (20)

(1 minus1198962

4Ω2119899

)119887119896minus119896V

2(119887119896minus2+ 119887119896+2) = 0 119896 = 3 5 7

(21)


119911 (119905) = 1198870 +

infin

sum119896=246

(119886119896sin 119896119905

2+ 119887119896cos 119896119905

2) (22)


(1 minus1

Ω2119899

)1198862minus119896V

21198864= 0 (23)

(1 minus1198962

4Ω2119899

)119886119896minus119896V

2(119886119896minus2+ 119886119896+2) = 0 119896 = 4 6 8

(24)

1198870minus119896V

21198872= 0

(1 minus1

Ω2119899

)1198872minus119896V

2(1198870+ 1198874) = 0

(25)

(1 minus1198962

4Ω2119899

)119887119896minus119896V

2(119887119896minus2+ 119887119896+2) = 0 119896 = 4 6 8

(26)


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 plusmn119896V

2minus1

4Ω2119899

minus119896V

20 sdot sdot sdot

minus119896V

21 minus

9

4Ω2119899

minus119896V

2sdot sdot sdot

0 minus119896V

21 minus

25

4Ω2119899

sdot sdot sdot

d

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (27)


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 minus1

Ω2119899

minus119896V

20 sdot sdot sdot

minus119896V

21 minus

4

Ω2119899

minus119896V

2sdot sdot sdot

0 minus119896V

21 minus

16

Ω2119899

sdot sdot sdot

d

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 minus119896V

20 0 sdot sdot sdot


Ω2119899

minus119896V

20 sdot sdot sdot

0 minus119896V

21 minus

4

Ω2119899

minus119896V

2sdot sdot sdot

0 0 minus119896V

21 minus

16

Ω2119899

sdot sdot sdot

d

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0

(28)











119889Φ

119889119905

10038161003816100381610038161003816100381610038161003816119911=120577=1


10158401015840

119905) 120577 (29)

119889119909

119889119905=120597Φ

120597119909

119889119911

119889119905=120597Φ

120597119911 (30)


Φ (119909 0 0) = minus119911119889119911119905(119905)

119889119905= 0

120577 (119909 0) = 1205770= 119886 cos (119896

119899119909)

(31)


119899= 119899Δ119905 (119899 = 0 1 2 3 ) at a


0 02 04 06 08 1 12 140

02

04

06

08

12

14

16

18

2

1


k = a1205962g

Ωn=120596n120596




120601 (119909 119911) =

119899

sum119895=1

119873119895(119909 119911) 120601

119895 (32)






intΩ

nabla119873119894

119898

sum119895=1

Φ119895nabla119873119895119889Ω|119895119894notinΓ

119904

= minusintΩ

nabla119873119894

119898

sum119895=1

120601119895nabla119873119895119889Ω|119895isinΓ

119904119894notinΓ119904

(33)



V = nabla119873119895sdot Φ119895 (34)


V = 1198861+ 1198862120585 + 1198863120578 + 1198864120585120578 (35)



4are unknowns which


119865 (119886) =

4

sum119894=1

V (120585119894 120578119894) minus V (120585

119894 120578119894)2 (36)


120597119865 (119886)

120597119886119896

= 0 119896 = 1 2 3 4 (37)



V = 119873 sdot V (38)



119904119894= (119909119894 119911119894 Φ119894) (39)

where119909119894= (1199091 1199092 119909

119873119883+1)119894

119911119894= (1199111 1199112 119911

119873119883+1)119894

Φ119894= (Φ1 Φ2 Φ

119873119883+1)119894

(40)


119863119904119894

119863119905= 119865 (119905

119894 119904119894) = 119865119894 (41)


119904119894+1= 119904119894+1199041

6+1199042

3+1199043

3+1199044

6 (42)

where1199041= nabla119905119865 (119905

119894 119904119894)

1199042= nabla119905119865 (119905

119894+nabla119905

2 119904119894+1199041

2)

1199043= nabla119905119865 (119905

119894+nabla119905

2 119904119894+1199042

2)

1199044= nabla119905119865 (119905

119894+ nabla119905 119904

119894+ 1199043)

(43)






119894 119910119894) (119894 = 1 2 119873119883+




119878119894=119894119871119891

119873119883 (44)


length 119878119894

(119909119894 119911119894) = 119891 (119878

119894) (45)















Case Ω119899

119896V 120596V 119886V1205962

V

1 1253 05 079811205961

05 g2 05 03 2120596

103 g

3 10 05 1205961

05 g4 05 02 2120596

302 g

5 06 05 1661205961

05 g6 055 05 18182120596

105 g

1

2

3

4

56

0 02 04 06 08 1 12 14k = a120596

2g

Ωn=120596n120596

02

04

06

08

12

14

16

18

2

1



1= 37594 rads and 120596

3=







0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(a)

0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(b)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(c)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(d)








0 10 20 30 40 50 60 70 80 90minus400

minus200

0

200

400

600

800

t1205961

120577(0h)a

(a)

minus400 minus200 0 200 400 600 800

minus800

minus600

minus400

minus200

0

200

400

600

120577a

120597120577120597ta1205961

(b)


phase-plane plot

t = 5 s

(a)

t = 10 s

(b)

t = 15 s

(c)

t = 18 s

(d)

t = 20 s

(e)

t = 21 s

(f)

t = 22 s

(g)

t = 23 s

(h)

t = 24 s

(i)



0 50 100 150 200 250 300 350 400minus150

minus100

minus50

0

50

100

150

200

250

t1205961

120577(0h)a

(a)

0 50 100 150 200 250 300minus100

minus50

0

50

100

150

120577(0h)a

t1205961

(b)


minus200

minus150

minus100

minus50

0

50

100

150

120577a

120597120577120597ta1205961

(c)

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

minus100 minus50 0 50 100 150120577a

120597120577120597ta1205961

(d)




1= 10 and forcing





t = 40 s

(a)

t = 65 s

(b)

t = 70 s

(c)

t = 75 s

(d)

t = 80 s

(e)

t = 82 s

(f)

t = 85 s

(g)

t = 88 s

(h)

t = 90 s

(i)




1= 06 and forcing







6 Conclusion



0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(a)

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(b)


0 10 20 30 40 50 60 70minus400

minus300

minus200

minus100

0

100

200

300

400

500

600

t1205961

120577(0h)a


1= 055 119896V = 05 and 120576 = 00014


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












the form

119865119903(119911) = 119890

119894119903119911119901 (119911) (16)




119899 119896V The plots


119911 (119905) =

infin

sum119896=135

(119886119896sin 119896119905

2+ 119887119896cos 119896119905

2) (17)


(1 +119896V

2minus1

4Ω2119899

)1198861minus119896V

21198863= 0 (18)

(1 minus1198962

4Ω2119899

)119886119896minus119896V

2(119886119896minus2+ 119886119896+2) = 0 119896 = 3 5 7

(19)

(1 minus119896V

2minus1

4Ω2119899

)1198871minus119896V

21198873= 0 (20)

(1 minus1198962

4Ω2119899

)119887119896minus119896V

2(119887119896minus2+ 119887119896+2) = 0 119896 = 3 5 7

(21)


119911 (119905) = 1198870 +

infin

sum119896=246

(119886119896sin 119896119905

2+ 119887119896cos 119896119905

2) (22)


(1 minus1

Ω2119899

)1198862minus119896V

21198864= 0 (23)

(1 minus1198962

4Ω2119899

)119886119896minus119896V

2(119886119896minus2+ 119886119896+2) = 0 119896 = 4 6 8

(24)

1198870minus119896V

21198872= 0

(1 minus1

Ω2119899

)1198872minus119896V

2(1198870+ 1198874) = 0

(25)

(1 minus1198962

4Ω2119899

)119887119896minus119896V

2(119887119896minus2+ 119887119896+2) = 0 119896 = 4 6 8

(26)


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 plusmn119896V

2minus1

4Ω2119899

minus119896V

20 sdot sdot sdot

minus119896V

21 minus

9

4Ω2119899

minus119896V

2sdot sdot sdot

0 minus119896V

21 minus

25

4Ω2119899

sdot sdot sdot

d

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (27)


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 minus1

Ω2119899

minus119896V

20 sdot sdot sdot

minus119896V

21 minus

4

Ω2119899

minus119896V

2sdot sdot sdot

0 minus119896V

21 minus

16

Ω2119899

sdot sdot sdot

d

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 minus119896V

20 0 sdot sdot sdot


Ω2119899

minus119896V

20 sdot sdot sdot

0 minus119896V

21 minus

4

Ω2119899

minus119896V

2sdot sdot sdot

0 0 minus119896V

21 minus

16

Ω2119899

sdot sdot sdot

d

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0

(28)











119889Φ

119889119905

10038161003816100381610038161003816100381610038161003816119911=120577=1


10158401015840

119905) 120577 (29)

119889119909

119889119905=120597Φ

120597119909

119889119911

119889119905=120597Φ

120597119911 (30)


Φ (119909 0 0) = minus119911119889119911119905(119905)

119889119905= 0

120577 (119909 0) = 1205770= 119886 cos (119896

119899119909)

(31)


119899= 119899Δ119905 (119899 = 0 1 2 3 ) at a


0 02 04 06 08 1 12 140

02

04

06

08

12

14

16

18

2

1


k = a1205962g

Ωn=120596n120596




120601 (119909 119911) =

119899

sum119895=1

119873119895(119909 119911) 120601

119895 (32)






intΩ

nabla119873119894

119898

sum119895=1

Φ119895nabla119873119895119889Ω|119895119894notinΓ

119904

= minusintΩ

nabla119873119894

119898

sum119895=1

120601119895nabla119873119895119889Ω|119895isinΓ

119904119894notinΓ119904

(33)



V = nabla119873119895sdot Φ119895 (34)


V = 1198861+ 1198862120585 + 1198863120578 + 1198864120585120578 (35)



4are unknowns which


119865 (119886) =

4

sum119894=1

V (120585119894 120578119894) minus V (120585

119894 120578119894)2 (36)


120597119865 (119886)

120597119886119896

= 0 119896 = 1 2 3 4 (37)



V = 119873 sdot V (38)



119904119894= (119909119894 119911119894 Φ119894) (39)

where119909119894= (1199091 1199092 119909

119873119883+1)119894

119911119894= (1199111 1199112 119911

119873119883+1)119894

Φ119894= (Φ1 Φ2 Φ

119873119883+1)119894

(40)


119863119904119894

119863119905= 119865 (119905

119894 119904119894) = 119865119894 (41)


119904119894+1= 119904119894+1199041

6+1199042

3+1199043

3+1199044

6 (42)

where1199041= nabla119905119865 (119905

119894 119904119894)

1199042= nabla119905119865 (119905

119894+nabla119905

2 119904119894+1199041

2)

1199043= nabla119905119865 (119905

119894+nabla119905

2 119904119894+1199042

2)

1199044= nabla119905119865 (119905

119894+ nabla119905 119904

119894+ 1199043)

(43)






119894 119910119894) (119894 = 1 2 119873119883+




119878119894=119894119871119891

119873119883 (44)


length 119878119894

(119909119894 119911119894) = 119891 (119878

119894) (45)















Case Ω119899

119896V 120596V 119886V1205962

V

1 1253 05 079811205961

05 g2 05 03 2120596

103 g

3 10 05 1205961

05 g4 05 02 2120596

302 g

5 06 05 1661205961

05 g6 055 05 18182120596

105 g

1

2

3

4

56

0 02 04 06 08 1 12 14k = a120596

2g

Ωn=120596n120596

02

04

06

08

12

14

16

18

2

1



1= 37594 rads and 120596

3=







0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(a)

0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(b)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(c)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(d)








0 10 20 30 40 50 60 70 80 90minus400

minus200

0

200

400

600

800

t1205961

120577(0h)a

(a)

minus400 minus200 0 200 400 600 800

minus800

minus600

minus400

minus200

0

200

400

600

120577a

120597120577120597ta1205961

(b)


phase-plane plot

t = 5 s

(a)

t = 10 s

(b)

t = 15 s

(c)

t = 18 s

(d)

t = 20 s

(e)

t = 21 s

(f)

t = 22 s

(g)

t = 23 s

(h)

t = 24 s

(i)



0 50 100 150 200 250 300 350 400minus150

minus100

minus50

0

50

100

150

200

250

t1205961

120577(0h)a

(a)

0 50 100 150 200 250 300minus100

minus50

0

50

100

150

120577(0h)a

t1205961

(b)


minus200

minus150

minus100

minus50

0

50

100

150

120577a

120597120577120597ta1205961

(c)

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

minus100 minus50 0 50 100 150120577a

120597120577120597ta1205961

(d)




1= 10 and forcing





t = 40 s

(a)

t = 65 s

(b)

t = 70 s

(c)

t = 75 s

(d)

t = 80 s

(e)

t = 82 s

(f)

t = 85 s

(g)

t = 88 s

(h)

t = 90 s

(i)




1= 06 and forcing







6 Conclusion



0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(a)

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(b)


0 10 20 30 40 50 60 70minus400

minus300

minus200

minus100

0

100

200

300

400

500

600

t1205961

120577(0h)a


1= 055 119896V = 05 and 120576 = 00014


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















119889Φ

119889119905

10038161003816100381610038161003816100381610038161003816119911=120577=1


10158401015840

119905) 120577 (29)

119889119909

119889119905=120597Φ

120597119909

119889119911

119889119905=120597Φ

120597119911 (30)


Φ (119909 0 0) = minus119911119889119911119905(119905)

119889119905= 0

120577 (119909 0) = 1205770= 119886 cos (119896

119899119909)

(31)


119899= 119899Δ119905 (119899 = 0 1 2 3 ) at a


0 02 04 06 08 1 12 140

02

04

06

08

12

14

16

18

2

1


k = a1205962g

Ωn=120596n120596




120601 (119909 119911) =

119899

sum119895=1

119873119895(119909 119911) 120601

119895 (32)






intΩ

nabla119873119894

119898

sum119895=1

Φ119895nabla119873119895119889Ω|119895119894notinΓ

119904

= minusintΩ

nabla119873119894

119898

sum119895=1

120601119895nabla119873119895119889Ω|119895isinΓ

119904119894notinΓ119904

(33)



V = nabla119873119895sdot Φ119895 (34)


V = 1198861+ 1198862120585 + 1198863120578 + 1198864120585120578 (35)



4are unknowns which


119865 (119886) =

4

sum119894=1

V (120585119894 120578119894) minus V (120585

119894 120578119894)2 (36)


120597119865 (119886)

120597119886119896

= 0 119896 = 1 2 3 4 (37)



V = 119873 sdot V (38)



119904119894= (119909119894 119911119894 Φ119894) (39)

where119909119894= (1199091 1199092 119909

119873119883+1)119894

119911119894= (1199111 1199112 119911

119873119883+1)119894

Φ119894= (Φ1 Φ2 Φ

119873119883+1)119894

(40)


119863119904119894

119863119905= 119865 (119905

119894 119904119894) = 119865119894 (41)


119904119894+1= 119904119894+1199041

6+1199042

3+1199043

3+1199044

6 (42)

where1199041= nabla119905119865 (119905

119894 119904119894)

1199042= nabla119905119865 (119905

119894+nabla119905

2 119904119894+1199041

2)

1199043= nabla119905119865 (119905

119894+nabla119905

2 119904119894+1199042

2)

1199044= nabla119905119865 (119905

119894+ nabla119905 119904

119894+ 1199043)

(43)






119894 119910119894) (119894 = 1 2 119873119883+




119878119894=119894119871119891

119873119883 (44)


length 119878119894

(119909119894 119911119894) = 119891 (119878

119894) (45)















Case Ω119899

119896V 120596V 119886V1205962

V

1 1253 05 079811205961

05 g2 05 03 2120596

103 g

3 10 05 1205961

05 g4 05 02 2120596

302 g

5 06 05 1661205961

05 g6 055 05 18182120596

105 g

1

2

3

4

56

0 02 04 06 08 1 12 14k = a120596

2g

Ωn=120596n120596

02

04

06

08

12

14

16

18

2

1



1= 37594 rads and 120596

3=







0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(a)

0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(b)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(c)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(d)








0 10 20 30 40 50 60 70 80 90minus400

minus200

0

200

400

600

800

t1205961

120577(0h)a

(a)

minus400 minus200 0 200 400 600 800

minus800

minus600

minus400

minus200

0

200

400

600

120577a

120597120577120597ta1205961

(b)


phase-plane plot

t = 5 s

(a)

t = 10 s

(b)

t = 15 s

(c)

t = 18 s

(d)

t = 20 s

(e)

t = 21 s

(f)

t = 22 s

(g)

t = 23 s

(h)

t = 24 s

(i)



0 50 100 150 200 250 300 350 400minus150

minus100

minus50

0

50

100

150

200

250

t1205961

120577(0h)a

(a)

0 50 100 150 200 250 300minus100

minus50

0

50

100

150

120577(0h)a

t1205961

(b)


minus200

minus150

minus100

minus50

0

50

100

150

120577a

120597120577120597ta1205961

(c)

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

minus100 minus50 0 50 100 150120577a

120597120577120597ta1205961

(d)




1= 10 and forcing





t = 40 s

(a)

t = 65 s

(b)

t = 70 s

(c)

t = 75 s

(d)

t = 80 s

(e)

t = 82 s

(f)

t = 85 s

(g)

t = 88 s

(h)

t = 90 s

(i)




1= 06 and forcing







6 Conclusion



0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(a)

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(b)


0 10 20 30 40 50 60 70minus400

minus300

minus200

minus100

0

100

200

300

400

500

600

t1205961

120577(0h)a


1= 055 119896V = 05 and 120576 = 00014


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












intΩ

nabla119873119894

119898

sum119895=1

Φ119895nabla119873119895119889Ω|119895119894notinΓ

119904

= minusintΩ

nabla119873119894

119898

sum119895=1

120601119895nabla119873119895119889Ω|119895isinΓ

119904119894notinΓ119904

(33)



V = nabla119873119895sdot Φ119895 (34)


V = 1198861+ 1198862120585 + 1198863120578 + 1198864120585120578 (35)



4are unknowns which


119865 (119886) =

4

sum119894=1

V (120585119894 120578119894) minus V (120585

119894 120578119894)2 (36)


120597119865 (119886)

120597119886119896

= 0 119896 = 1 2 3 4 (37)



V = 119873 sdot V (38)



119904119894= (119909119894 119911119894 Φ119894) (39)

where119909119894= (1199091 1199092 119909

119873119883+1)119894

119911119894= (1199111 1199112 119911

119873119883+1)119894

Φ119894= (Φ1 Φ2 Φ

119873119883+1)119894

(40)


119863119904119894

119863119905= 119865 (119905

119894 119904119894) = 119865119894 (41)


119904119894+1= 119904119894+1199041

6+1199042

3+1199043

3+1199044

6 (42)

where1199041= nabla119905119865 (119905

119894 119904119894)

1199042= nabla119905119865 (119905

119894+nabla119905

2 119904119894+1199041

2)

1199043= nabla119905119865 (119905

119894+nabla119905

2 119904119894+1199042

2)

1199044= nabla119905119865 (119905

119894+ nabla119905 119904

119894+ 1199043)

(43)






119894 119910119894) (119894 = 1 2 119873119883+




119878119894=119894119871119891

119873119883 (44)


length 119878119894

(119909119894 119911119894) = 119891 (119878

119894) (45)















Case Ω119899

119896V 120596V 119886V1205962

V

1 1253 05 079811205961

05 g2 05 03 2120596

103 g

3 10 05 1205961

05 g4 05 02 2120596

302 g

5 06 05 1661205961

05 g6 055 05 18182120596

105 g

1

2

3

4

56

0 02 04 06 08 1 12 14k = a120596

2g

Ωn=120596n120596

02

04

06

08

12

14

16

18

2

1



1= 37594 rads and 120596

3=







0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(a)

0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(b)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(c)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(d)








0 10 20 30 40 50 60 70 80 90minus400

minus200

0

200

400

600

800

t1205961

120577(0h)a

(a)

minus400 minus200 0 200 400 600 800

minus800

minus600

minus400

minus200

0

200

400

600

120577a

120597120577120597ta1205961

(b)


phase-plane plot

t = 5 s

(a)

t = 10 s

(b)

t = 15 s

(c)

t = 18 s

(d)

t = 20 s

(e)

t = 21 s

(f)

t = 22 s

(g)

t = 23 s

(h)

t = 24 s

(i)



0 50 100 150 200 250 300 350 400minus150

minus100

minus50

0

50

100

150

200

250

t1205961

120577(0h)a

(a)

0 50 100 150 200 250 300minus100

minus50

0

50

100

150

120577(0h)a

t1205961

(b)


minus200

minus150

minus100

minus50

0

50

100

150

120577a

120597120577120597ta1205961

(c)

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

minus100 minus50 0 50 100 150120577a

120597120577120597ta1205961

(d)




1= 10 and forcing





t = 40 s

(a)

t = 65 s

(b)

t = 70 s

(c)

t = 75 s

(d)

t = 80 s

(e)

t = 82 s

(f)

t = 85 s

(g)

t = 88 s

(h)

t = 90 s

(i)




1= 06 and forcing







6 Conclusion



0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(a)

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(b)


0 10 20 30 40 50 60 70minus400

minus300

minus200

minus100

0

100

200

300

400

500

600

t1205961

120577(0h)a


1= 055 119896V = 05 and 120576 = 00014


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119894 119910119894) (119894 = 1 2 119873119883+




119878119894=119894119871119891

119873119883 (44)


length 119878119894

(119909119894 119911119894) = 119891 (119878

119894) (45)















Case Ω119899

119896V 120596V 119886V1205962

V

1 1253 05 079811205961

05 g2 05 03 2120596

103 g

3 10 05 1205961

05 g4 05 02 2120596

302 g

5 06 05 1661205961

05 g6 055 05 18182120596

105 g

1

2

3

4

56

0 02 04 06 08 1 12 14k = a120596

2g

Ωn=120596n120596

02

04

06

08

12

14

16

18

2

1



1= 37594 rads and 120596

3=







0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(a)

0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(b)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(c)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(d)








0 10 20 30 40 50 60 70 80 90minus400

minus200

0

200

400

600

800

t1205961

120577(0h)a

(a)

minus400 minus200 0 200 400 600 800

minus800

minus600

minus400

minus200

0

200

400

600

120577a

120597120577120597ta1205961

(b)


phase-plane plot

t = 5 s

(a)

t = 10 s

(b)

t = 15 s

(c)

t = 18 s

(d)

t = 20 s

(e)

t = 21 s

(f)

t = 22 s

(g)

t = 23 s

(h)

t = 24 s

(i)



0 50 100 150 200 250 300 350 400minus150

minus100

minus50

0

50

100

150

200

250

t1205961

120577(0h)a

(a)

0 50 100 150 200 250 300minus100

minus50

0

50

100

150

120577(0h)a

t1205961

(b)


minus200

minus150

minus100

minus50

0

50

100

150

120577a

120597120577120597ta1205961

(c)

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

minus100 minus50 0 50 100 150120577a

120597120577120597ta1205961

(d)




1= 10 and forcing





t = 40 s

(a)

t = 65 s

(b)

t = 70 s

(c)

t = 75 s

(d)

t = 80 s

(e)

t = 82 s

(f)

t = 85 s

(g)

t = 88 s

(h)

t = 90 s

(i)




1= 06 and forcing







6 Conclusion



0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(a)

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(b)


0 10 20 30 40 50 60 70minus400

minus300

minus200

minus100

0

100

200

300

400

500

600

t1205961

120577(0h)a


1= 055 119896V = 05 and 120576 = 00014


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(a)

0 20 40 60 80 100minus2

minus15

minus1

minus05

0

05

1

15

2

PresentFrandsen

t1205961

120577(0h)a

(b)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(c)


minus15

minus1

minus05

0

05

1

15

2

120577a

120597120577120597ta1205961

(d)








0 10 20 30 40 50 60 70 80 90minus400

minus200

0

200

400

600

800

t1205961

120577(0h)a

(a)

minus400 minus200 0 200 400 600 800

minus800

minus600

minus400

minus200

0

200

400

600

120577a

120597120577120597ta1205961

(b)


phase-plane plot

t = 5 s

(a)

t = 10 s

(b)

t = 15 s

(c)

t = 18 s

(d)

t = 20 s

(e)

t = 21 s

(f)

t = 22 s

(g)

t = 23 s

(h)

t = 24 s

(i)



0 50 100 150 200 250 300 350 400minus150

minus100

minus50

0

50

100

150

200

250

t1205961

120577(0h)a

(a)

0 50 100 150 200 250 300minus100

minus50

0

50

100

150

120577(0h)a

t1205961

(b)


minus200

minus150

minus100

minus50

0

50

100

150

120577a

120597120577120597ta1205961

(c)

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

minus100 minus50 0 50 100 150120577a

120597120577120597ta1205961

(d)




1= 10 and forcing





t = 40 s

(a)

t = 65 s

(b)

t = 70 s

(c)

t = 75 s

(d)

t = 80 s

(e)

t = 82 s

(f)

t = 85 s

(g)

t = 88 s

(h)

t = 90 s

(i)




1= 06 and forcing







6 Conclusion



0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(a)

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(b)


0 10 20 30 40 50 60 70minus400

minus300

minus200

minus100

0

100

200

300

400

500

600

t1205961

120577(0h)a


1= 055 119896V = 05 and 120576 = 00014


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 10 20 30 40 50 60 70 80 90minus400

minus200

0

200

400

600

800

t1205961

120577(0h)a

(a)

minus400 minus200 0 200 400 600 800

minus800

minus600

minus400

minus200

0

200

400

600

120577a

120597120577120597ta1205961

(b)


phase-plane plot

t = 5 s

(a)

t = 10 s

(b)

t = 15 s

(c)

t = 18 s

(d)

t = 20 s

(e)

t = 21 s

(f)

t = 22 s

(g)

t = 23 s

(h)

t = 24 s

(i)



0 50 100 150 200 250 300 350 400minus150

minus100

minus50

0

50

100

150

200

250

t1205961

120577(0h)a

(a)

0 50 100 150 200 250 300minus100

minus50

0

50

100

150

120577(0h)a

t1205961

(b)


minus200

minus150

minus100

minus50

0

50

100

150

120577a

120597120577120597ta1205961

(c)

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

minus100 minus50 0 50 100 150120577a

120597120577120597ta1205961

(d)




1= 10 and forcing





t = 40 s

(a)

t = 65 s

(b)

t = 70 s

(c)

t = 75 s

(d)

t = 80 s

(e)

t = 82 s

(f)

t = 85 s

(g)

t = 88 s

(h)

t = 90 s

(i)




1= 06 and forcing







6 Conclusion



0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(a)

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(b)


0 10 20 30 40 50 60 70minus400

minus300

minus200

minus100

0

100

200

300

400

500

600

t1205961

120577(0h)a


1= 055 119896V = 05 and 120576 = 00014


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 50 100 150 200 250 300 350 400minus150

minus100

minus50

0

50

100

150

200

250

t1205961

120577(0h)a

(a)

0 50 100 150 200 250 300minus100

minus50

0

50

100

150

120577(0h)a

t1205961

(b)


minus200

minus150

minus100

minus50

0

50

100

150

120577a

120597120577120597ta1205961

(c)

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

minus100 minus50 0 50 100 150120577a

120597120577120597ta1205961

(d)




1= 10 and forcing





t = 40 s

(a)

t = 65 s

(b)

t = 70 s

(c)

t = 75 s

(d)

t = 80 s

(e)

t = 82 s

(f)

t = 85 s

(g)

t = 88 s

(h)

t = 90 s

(i)




1= 06 and forcing







6 Conclusion



0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(a)

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(b)


0 10 20 30 40 50 60 70minus400

minus300

minus200

minus100

0

100

200

300

400

500

600

t1205961

120577(0h)a


1= 055 119896V = 05 and 120576 = 00014


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










t = 40 s

(a)

t = 65 s

(b)

t = 70 s

(c)

t = 75 s

(d)

t = 80 s

(e)

t = 82 s

(f)

t = 85 s

(g)

t = 88 s

(h)

t = 90 s

(i)




1= 06 and forcing







6 Conclusion



0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(a)

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(b)


0 10 20 30 40 50 60 70minus400

minus300

minus200

minus100

0

100

200

300

400

500

600

t1205961

120577(0h)a


1= 055 119896V = 05 and 120576 = 00014


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(a)

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

t1205961

120577(0h)a

PresentFrandsen

(b)


0 10 20 30 40 50 60 70minus400

minus300

minus200

minus100

0

100

200

300

400

500

600

t1205961

120577(0h)a


1= 055 119896V = 05 and 120576 = 00014


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Finite Element Simulation of Dynamic...

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