The dynamics and stability of pipes conveying fluid
has been studied thoroughly in the last decades. In this
paper we study the stability in the Liapunov sense, of
a clamped-pinned pipe conveying fluid at a low speed.
After describing the motion of the system by partial
differential equations we solve equations using finite
element method testing solutions by means of ANSYS,
we analyze the characteristic equation and its eigenvalues
in order to obtain the stability conditionsPreprin

García Planas, María Isabel

Mediano Valiente, Begoña

English

UPCommons. Portal del coneixement obert de la UPC

PHYSCON 2011, Leo´n, Spain, September, 5–September, 8 2011STABILITY OF A PIPELINE HYDRAULIC FLUID WITHONE END FIXEDB. Mediano-ValienteDepartamento de Matema`tica Aplicada IUniversitat Polite`cnica de CatalunyaSpainbego med@yahoo.esM. I. Garcı´a-PlanasDepartament de Matema`tica Aplicada IUniversitat Polite`cnica de CatalunyaSpainmaria.isabel.garcia@upc.eduAbstractThe dynamics and stability of pipes conveying fluidhas been studied thoroughly in the last decades. In thispaper we study the stability in the Liapunov sense, ofa clamped-pinned pipe conveying fluid at a low speed.After describing the motion of the system by partialdifferential equations we solve equations using finiteelement method testing solutions by means of ANSYS,we analyze the characteristic equation and its eigenval-ues in order to obtain the stability conditions.Key wordsStability, eigenvalues, pipe conveying fluid.1 IntroductionThe dynamics and stability of pipes conveying fluidhas been studied thoroughly in the last decades see forexample [1; 2; 3].It is well known that the dynamical behaviour of pipesof a finite length depends strongly on the type of bound-ary. We must distinguish between the type of supports(fixed, one end fixed, etc.) and their location (horizon-tal, vertical). In this paper we refer to a one end fixedhorizontal pipeline, respectively.The dynamics of the system can be described by a par-tial differential equation [4; 5]EI ∂4y∂x4 + (mfU2 − T ) ∂2y∂x2+2mfU∂2y∂x∂t + (mp +mf )∂2y∂t2 = 0(1)Considering boundary conditions at ends of a clamped-pinned pipe we find approximate solution usingGalerkin’s methode obtaining as a result a linear gy-roscopic system possessing the properties of linearHamiltonian systems.We compute the eigenvalues of this linear Hamilto-nian system in order to study the stability. It is knownthat the stability of a linear Hamiltonian system is notFigure 1. Pipelineasymptotic, nevertheless the study provides the neces-sary stability condition for the original non-linear sys-temThe paper is structured as follows. Section 2 presentsa mathematical statement of the problem. Section 3is devoted to analyze the stability of linear gyroscopicsystem obtained in section 1. Section 4 presents a sim-ulation of the dynamic system using ANSYS.2 PreliminariesThe system under consideration is a straight, tight andof finite length pipeline, passing through it a fluid. Thefollowing assumptions are taken into account in theanalysis of the system:i) Are ignored the effects of gravity, the coefficientdamping material, the shear strain and rotationalinertiaii) The pipeline is considered horizontaliii) The pipe is inextensibleiv) The lateral movement of y(x, t) is small, and withlarge length wave compared with the diameter ofthe pipe, so that theory Euler-Bernoulli is applica-ble for the description of vibration bending of thepipe.v) It ignores the velocity distribution in the cross sec-tion of pipe.The equation for a single span prestressed pipelinewhere the fluid is transported is a function of the dis-tance x and time t and is based on the beam theory:EI∂4y∂x4+mp∂2y∂x2= fint(x, t) (2)where EI is the bending stiffness of the pipe (Nm2),mp is the pipe mass per unit length (kgm ) and fint is aninside force acting on the pipe.The internal fluid flow is approximated as a plug flow,so all points of the fluid have the same velocity U rela-tive to the pipe. This is a reasonable approximation fora turbulent flow profile. Because of that we cant writethe inside force as:fint = −mf d2ydt2∣∣∣∣x=Ut(3)where mf is the fluid mass per unit length (kgm ) and Uis the fluid velocity (ms ).The internal fluid causes an hydrostatic pressure onthe pipe wall.T = −AiPi (4)where Ai is the internal cross sectional area of the pipe(m2) and Pi is the hydrostatic pressure inside the pipe(Pa).Finally if we consider that the total acceleration isequal to the composition of local, carioles and centrifu-gal acceleration. The resulting equation is (1):EI ∂4y∂x4 + (mfU2 − T ) ∂2y∂x2+2mfU∂2y∂x∂t + (mp +mf )∂2y∂t2 = 0(5)The boundary conditions at ends of a clamped-pinnedpipe are given as:y(0, t) = 0, y(L, t) = 0∂2y(0, t)∂t= 0, EI∂2y(L, t)∂x2= −Krs∂y(L, t)∂x(6)where Krs is the stiffness of the rotational spring at theright end.Applying Galerkin method and taking n = 2 the ap-proximate solution is:Figure 2. Boundary conditionsy(x, t) = q1(t)senpiLx+ q2(t)sen2piLxReplacing the solution in the equation 5, we get:EIq1(t)(pi4L4 senpiLx+ q2(t)16pi4L4 sen2piL x)+(mfU2 − T )(−q1(t) pi2L2 senpiLx− q2(t)4pi2L2 sen2piL x)+2mfU(q˙1(t)piL cospiLx+ q˙2(t)2piL cos2piL x)+(mp +mf)(q¨1(t)senpiLx+ q¨2(t)sen2piL x)= 0(7)Multiplying by sen piLξ and sen2piL ξ, respectively, weobtain:L2 (mp +mf )q¨1(t)− 83mfUq˙2(t)+(EI pi42L3 −(mfU2−T )pi22L)q1(t) = 0L2 (mp +mf )q¨2(t)− 83mfUq˙1(t)+(EI 8pi4L3 − (mfU2 − T )4pi2L2)q2(t) = 0(8)The previous equation system can be written as matrixform like:Mq¨ +Bq˙ + Cq = 0Where gyroscopic lineal system is:x¨+Gx˙+Kx = 0withG = M−1/2BM−1/2, K = M−1/2CM−1/2.Applied to our study is:M−1/2 =1√L2 (mp +mf)(1 00 1)so, it remains:G =16mfL(mf +mp)(0 −11 0)K =2L(mf +mp)(K1 00 K2)with:K1 = EIpi42L3− (mfU2 − T )pi22LK2 = EI8pi4L3− (mfU2 − T )4pi2L2Introducing the vector:(xy)=(xx˙+Gx/2)and calculating the derivatives of x and y we found x˙ =y − Gx/2, y˙ = x¨ + Gx˙/2 and considering that x¨ =−Gx˙−Kx linearizing the system we get:(x˙y˙)=( −G/2 I2G2/4−K −G/2)(xy)The matrix A of this system is Hamiltonian becauseQA is symmetrical, where Q is antisymmetrical:Q =0 0 1 00 0 0 1−1 0 0 00 −1 0 0If we consider the following parameters:Λ =EIpi4L3δ = (mfU2 − T )pi2Lβ =1L(mf +mp)(9)the matrix G and K are written as:G = 16mfβ(0 −11 0)K = 2β(12Λ− 12δ 00 8Λ− 4Lδ)(10)Therefore, matrix A is:A =0 a 1 0−a 0 0 1b 0 0 a0 c −a 0 (11)where:a = 8mfβb = −64m2fβ2 − βΛ + βδc = −64m2fβ2 − 16βΛ + 8Lβδ(12)The characteristic equation of the matrix is:λ4 + (2a2 − b− c)λ2 + (a2 + c)(a2 + b) = 0 (13)that is a biquadratic equation, where we find the eigen-values of the system:λ = ±√−2a2 + b+ c±√−8a2b− 8a2c− 2bc+ b2 + c22(14)If we change the previous equation with the parame-ters, we extract:λ4 +(256m2fβ2 + 17βΛ−(1 +8L)βδ)λ2+16β2Λ2 −(16 +8L)β2Λδ +8Lβ2δ2 = 0 (15)λ = ±√λ1 ± β√λ22(16)withλ1 = −256m2fβ2 − 17βΛ +(1 + 8L)βδλ2 = 65536m4fβ2 + 8704m2fβΛ−(512 + 4096L)m2fβδ+225Λ2 +(1 + 64L2− 16L)δ2 +(30− 240L)Λδ(17)3 StabilityIn this section we study the stability properties of lin-ear dynamic systems representing the pipeline. Also,we will present a detailed explanation of the effect ofthe stabilization in terms of the bifurcation theory ofeigenvalues.Eigenvalues of the matrix (11) characterize the stabil-ity of the Hamiltonian system. The system is stable ifthe eigenvalues lyes on the imaginary axe.Taking into account that the values in the system areknow only approximately, the matrix A in the systemcan be considered as a family of matrices depending onparameters a, b, c in a neighborhood of a fixed point p0,that permit us to study the stability border.Stability conditions requires that the roots obtained in(16), λ2 = λ1 ± β√λ22are real and negative. Impos-ing these conditions we can determine the stability zonein the parameter space.We observe that the points p = (a, b, c) such that2a2 − b− c = 0(a2 + c)(a2 + b) = 0}, (18)the characteristic polynomial is λ4,If a 6= 0 the Jordan form of A is0 1 0 00 0 1 00 0 0 10 0 0 0If a = 0 the Jordan form of A is0 1 0 00 0 1 00 0 0 00 0 0 0At these points we have singularities of the type 04 y03 on the stability boundary.Also we obtain the eigenvalue 0 at the points such that(a, b, c)(a2 + c)(a2 + b) = 02a2 − b− c 6= 0}(19)At the points (a, b,−a2) we have two possibilities de-pending on b if it is equal or not to −a2For b 6= −a2 the Jordan form is0 1 0 00 0 0 00 0√3a2 − b 00 0 0 −√3a2 − bFor b = −a2 the Jordan form is0 0 0 00 0 0 00 0√3a2 − b 00 0 0 −√3a2 − b.To ensure stability we need the non-zero eigenvalueslye in the imaginary axe we have 3a2 − b < 0, so b =−a2 is out of stability space.At the points (a,−a2, c) we have two cases dependingon c be equal or not to −a2For c 6= −a2 the Jordan form is0 1 0 00 0 0 00 0√3a2 − c 00 0 0 −√3a2 − cFor c = −a2 the Jordan form is0 0 0 00 0 0 00 0√3a2 − c 00 0 0 −√3a2 − cAnalogously, the case c = a2 is out of the stabilityspaceFor the case b 6= −a2 and c 6= −a2 we have singular-ities of the type 02 in the boundary of stability.It remains to study the case that no eigenvalue is zeroThe roots of µ2+(2a2−b−c)µ+(a2+c)(a2+b) = 0,are real and negative when2a2 − b− c > 0(a2 + c)(a2 + b) > 0(2a2 − b− c)2 ≥ 4(a2 + c)(a2 + b) (20)In the case (2a2 − b − c)2 = 4(a2 + c)(a2 + b) theeigenvalues are λ = ±i√2a2 − b− c = ±iω double.It is easy to observe that rank (A − (±iω)I) = 3 sothe Jordan form isiω 1 0 00 iω 0 00 0 −iω 10 0 0 −iωAt the points (a, b, c) with 2a2 − b − c > 0, (a2 +c)(a2+ b) > 0 and (2a2− b− c)2 = 4(a2+ c)(a2+ b)we have singularities of the type ±iω2.The last case2a2 − b− c > 0(a2 + c)(a2 + b) > 0(2a2 − b− c)2 > 4(a2 + c)(a2 + b) (21)determined the stability points (a, b, c) remainingwithin the area bounded by the above singularities.Removing the variable change we know that:a =8mfL(mf+mp)b =−64m2fL2(mf+mp)2 − EIpi4L4(mf+mp)+(mfU2+AiPi)pi2L2(mf+mp)c =−64m2fL2(mf+mp)2 − 16EIpi4L4(mf+mp)+8(mfU2+AiPi)pi2L3(mf+mp)Taking as a constant parameters L = 1000mm, I =2, 185 · 106, Ai = 2500pi due to the geometry of thepipe and mf = 2, 5pi · 10−6 Tnmm assuming the fluid aswater. We also suppose that the study is applied to theinside wall of the pipe so U at these points are zero.Therefore the values a, b y c are:a = 2pi·10−8(2,5pi·10−6+mp)b = −4·10−16pi2(2,5pi·10−6+mp)2−2,185·10−6Epi4(2,5pi·10−6+mp)+2,5·10−3Pipi3(2,5pi·10−6+mp)c = −4·10−16pi2(2,5pi·10−6+mp)2−34,96·10−6Epi4(2,5pi·10−6+mp)+2·10−5Pipi3(2,5pi·10−6+mp)That permit us to obtain the following relation de-pending only on mp, E and Pi:16 · 10−132.5 · 10−6pi +mp + 37.145 · 10−3pi2E − 2.52piPi > 015.27752 · 10−4pi2E2 + P 2i − 17.48874 · 10−1piEPi > 0(16 · 10−132.5 · 10−6pi +mp + 37.145 · 10−3piE − 2.52Pi)2>4pi2(76.3877 · 10−6pi2E2 − 87.4437 · 10−3piEPi+5 · 10−2P 2i )(22)We make this study to show the stability of pipes withdifferent materials assuming in all of them that the fluidtransported is water and causes a constant pressure onits walls of 4 bar. The geometrical conditions of thepipe are the inside diameter equal to 50 mm and thethickness of the pipe which is 6 mm. The materialschosen are PVC, Polyethylene and Concrete.The values of E and mp of the PVC pipe are:E = 30, 581Nmm2mp = 2, 76 · 10−6 TnmmApplying the inequalities (22) we found that the solu-tion is unstable.The values of E and mp of the PE pipe are:E = 9, 174Nmm2mp = 1, 91 · 10−6 TnmmApplying the inequalities (22) we found that the solu-tion is unstable.The values of E and mp of the Concrete pipe are:E = 221, 203Nmm2mp = 4, 40 · 10−6 TnmmApplying the inequalities (22) we found that the solu-tion is stable.We observe that the case of PVC pipe is the furthestaway from stability zone.4 SimulationTo demonstrate the stabilities found in the previouschapter we calculate vibration characteristics of a pipeconveying fluid using a Finite Element package calledANSYS. To determinate the vibration characteristicswe used modal analysis, with this analysis you findnatural frequencies and mode shapes which are impor-tant parameters in the design of a structure for dynamicstudies.In the following pictures it is shown the performanceof the first and the second shapes and the natural fre-quencies of them.Figure 3. First shape of PVCFigure 4. First shape of PolyethyleneAs seen in picture 3, 4 and 5 the lowest natural fre-quency is the concrete pipe (it is not write on the picturebecause is zero) and the biggest one is the Polyethylenepipe (19, 435Hz) but the greater displacement of x axisis the PVC pipe. This combination result in instabil-ity of Polyethylene and PVC pipe whereas in Concretepipe is stable.Figure 5. First shape of ConcreteFigure 6. Second shape of PVCAs seen in picture 6, 7 and 8 the lowest natural fre-quency is the concrete pipe (0,0238 Hz) and the biggestone is the PVC pipe (24, 043 Hz) but the greater dis-placement of x axis is the PE pipe. This combina-tion result in instability of Polyethylene and PVC pipewhereas in Concrete pipe is stable.Figure 7. Second shape of PolyethyleneFigure 8. Second shape of Concrete5 ConclusionIn this paper we have compared the calculations andthe simulation of typical materials for a pipe used inpublic works.We have shown that the dynamics and stability ofpipes conveying fluid not only depends on the boundaryconditions but it is also strongly important the materialof the pipe and the pressure produced by the fluid.ReferencesG.L. Kuiper and A.V. Metrikine, On stability of aclamped-pinned pipe conveying fluid Heron, 49, (3),pp. 211-232, (2004).A. K. Misra, S. S. T. Wong, M. P. Paı¨doussis, Dy-namics and stability of pinned-clamped and clamped-pinned cylindrical shells conveying fluid. Journal ofFluids and Structures 15, (8), pp. 1153-1166, (2001).Paı¨doussis, M.P., Tian, B., Misra, A.K., The dy-namic and stability of pinned-clamped coaxial cylin-drical shells conveying viscous flow. Proceedings ofthe Canadian Congress of Applied Mechanics, CAN-CAM 93, pp. 259-260, (1993).A. P. Seyranian, A.A. Mailybaev, “Multiparame-ter Stability Theory with Mechanical Applications”World Scientific, Singapore, 2003.J. M. T. Thompson, “Instabilities and Catastrophes inScience and Engineering” Wiley, New York, 1982.

Stability of a pipeline hydraulic fluid with one end fixed

The dynamics and stability of pipes conveying fluid

has been studied thoroughly in the last decades. In this

paper we study the stability in the Liapunov sense, of

a clamped-pinned pipe conveying fluid at a low speed.

After describing the motion of the system by partial

differential equations we solve equations using finite

element method testing solutions by means of ANSYS,

we analyze the characteristic equation and its eigenvalues

in order to obtain the stability condition

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https://upcommons.upc.edu/bitstream/2117/18166/1/bmediano.pdf

Stability of a pipeline hydraulic fluid with one end fixed

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