Available online at www.sciencedirect.com Procedia Engineering () 986 99 The Twelfth East Asia-Pacific Conference on Structural Engineering and Construction Flow-Induced Vibration of Pipeline on Elastic Support a R. T. FAAL, b D. DERAKHSHAN FACULTY OF ENGINEERING, ZANJAN UNIVERSITY, ZANJAN, IRAN Abstract We study the self-excited flexural vibration of a pipe due to internal flow which is entirely supported on elastic foundation. We consider a finite length pipe with clamped ends made of isotropic material consist of an internal flow with constant velocity. The Euler-Bernoulli model of beam is considered to analysis the vibratory behavior of pipe. The governing equation of motion is then derived which is partial differential equation in terms of derivative of transverse displacement of pipe with respect to time and axial distance. The full term governing equation of motion contains the second derivative of transverse displacement with respect to time and axial distance which has been neglected in earlier work is considered here and solved analytically. The natural frequencies of coupled pipe-fluid system are then obtained and the effects of the neglected term on the natural frequencies of system are studied. The effects of the stiffness of elastic foundation, velocity and density of inner fluid, inner diameter of pipe with constant thickness, elasticity modulus of the pipe and finally pipe length are studied. The stability analysis of the vibration is also accomplished. Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and/or peer-review under responsibility of [name organizer] Keywords: Flow-induced Vibration, elastic foundation, flexural vibration INTRODUCTION Concern for the flow-induced vibration of the tube has become a serious consideration in the design of shell-and-tube equipment. All tubes vibrate in all flow conditions! However, we are concerned with the vibrations that cause significant damage to the pipe. Structural failure due to flow-induced vibration is a common problem of heat exchangers which affects on the reliability and performance of them. Also flowinduced vibrations can damage the tubes in evaporators. On the other hand because of increasing urban a Corresponding author: Email: faal9@yahoo.com b Presenter: faal9@yahoo.com 877 758 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:.6/j.proeng..7.76
R.T. FAAL and D. DERAKHSHAN / Procedia Engineering () 986 99 987 population, under soft ground piping is an increasingly common activity to manage oil or water transportation facilities in many large cities around the world. In this regards determination of flowinduced lateral vibration of buried pipelines due to an internal flow become a subject of considerable importance in this field. Literature is replete with studies related to flow-induced vibrations in the last four decades. The first attempt in flow-induced vibration study was done by (Ashley and Haviland 95). The practical vibration problems of heat exchanger tubes in power plants and nuclear reactors were the subject of many studies as other aspects of this field. The finite elements models and also other numerical methods are frequently used for these studies. For want of space, we review the studies which are only based on the analytical methods. Most of the earlier studies are related to investigations about inner transient flow with time-varying velocity, which are reviewed at first here. Coupled pipe-dynamics equations for the axial, radial, and lateral vibrations of the pipeline as well as for the transients of unsteady internal fluid pressure and velocity were the subject of studies by (Lee and Kim 999) and (Lee et al. 995). Afterward, (Gorman et al. ) included the effects of radial shell vibration and initial axial tension on Lee s pipe-dynamics model. A spectral element model is developed by (Lee and Park 6) for the uniform straight pipelines with an internal unsteady fluid. Four coupled pipe-dynamics equations are derived in terms of the transverse displacement, the axial displacement, the fluid pressure and velocity and then linearized them. The fast Fourier transform FFT is used to investigate the structural dynamic characteristics and the internal fluid transient properties. The wave characteristics, divergence stability and dynamics of the viscoelastic pipelines conveying internal flow are examined by (Lee et al. 9) using the spectral element method. Small perturbations with respect to the steady state values of inner fluid velocity and pressure is considered to make the governing equations to be linear. In the spectral element model the governing differential equation of motion is transformed into the frequency-domain by using the discrete Fourier transformation theory. The internal flow velocity at which the divergence instability occurs is derived in an analytical form. In second part of review we consider steady-state flow-induced vibration problem in pipe conveying an internal flow with constant velocity. The self-excited flexural vibration of a pipe due to an internal flow with constant velocity was investigated by (Biswas and Ahmed ). In the above-mentioned paper the second order derivatives of transverse displacement with respect to time and axial distance (product term) is neglected in the governing equation. The critical value of fluid velocity as a stability boundary is evaluated in terms of the system parameters such as, pipe material density and rigidity and fluid density, and so on. The optimal flow velocity for minimum vibration was obtained. In this study the full term governing equation of motion is solved analytically i.e. the first derivative of transverse displacement with respect to time and axial distance has not been neglected and the weaknesses of the earlier work is studied. To derive this governing equation, an infinitesimal element of inner fluid of pipe is chosen which is imposed to horizontal movement with constant velocity and lateral movement due to pipe flexural vibration. Therefore the transverse acceleration of this element may be evaluated using concept of total derivative or material derivative of transverse displacement of the pipe which contains time derivative and convective inertia term. Consequently, the distributed force per unit length of the pipe is the summation of inertia force of the fluid, the inertia force of the pipe and also the distributed force of the elastic foundation. Applying the boundary conditions, the natural frequencies of coupled pipe-fluid system are then obtained. Analysis of the effects of stiffness of the elastic foundation and also the complete governing equation on the natural frequencies of the coupled system and also analytic solution of problem may be the novelty of the present work.
988 R.T. FAAL and D. DERAKHSHAN / Procedia Engineering () 986 99 DERIVATION OF GOVERNING EQUATION OF PIPELINE ON ELASTIC SUPPORT We consider an elastic straight pipe with length and two clamped ends which is totally bonded on elastic support with the stiffness per unit length k. The bending rigidity and the inner surface of pipe are EI and A, respectively. The pipe contains an incompressible fluid with density f and axial velocity U. Each element of inner fluid undergoes two movements consist of the horizontal movement with the velocityu and the vertical displacement Y ( X, t) (where X is the axial distance from one end) resulting of the pipe flexural vibration, therefore the acceleration of this element may be easily written as a( X, t) ( U ) Y ( X, t) X t U X t is the total derivative or material derivative operator. Hence the flow induced vertical force on the unit length of pipe is F( X, t) f Aa( X, t). Considering the Euler-Bernoulli beam theory; the governing equation of flexural vibration of pipe in view of Eq. () and relation F( X, t) Aa( X, t) is attained as where Y EI X ( m f mp Y f A( U ) t X t Y ky where m is the mass per unit length of the pipe. The above equation is simplified as p p Y Y f A) AU AU f f t Xt Y X Y EI X ky The transformation x X /, y Y / and t / a are used to normalize the above equation in convenient form (dimensionless form) where a mp / EI is a parameter with the dimension of the time and all others are dimensionless. Making use of above transformation, the Eq. () may be rewritten as follows y y ( ) u ( ) u x y y by x x where b k / EI, u au / and mp /( mp f A) are dimensionless parameters. The ends boundary conditions imply that y(,),y(,) y y (,), (,) x x i( trx ) There is a conventional harmonic solution of Eq. () in the form of Y ( X, t) Y ( X ) e which is usual solution in the wave motion problems which is the circular frequency and is r a real constant. i( x) Consequently the normalized solution of Eq. () may be considered in the form of y( x, ) y ( x) e where a and r. Substituting this solution form into Eq. () and setting two the real and imaginary parts of the ensuing equation equal to zero results in () () () () (5) y () ( x) [( ) u 6 ] y ( x) [ ( ) u y ( x) [ u ( )( u) ] y ( x) b u( )] y ( x) (6)
R.T. FAAL and D. DERAKHSHAN / Procedia Engineering () 986 99 989 The characteristic equations of above linear ordinary differential equations in terms of s are s s [( ) u 6 ] s [ [ u ( )( u) ] s ( ) u b u( )] (7) The condition for the existence of the solution for the system of ordinary differential equations (6) is the existence of identical roots for two algebraic equations (7). From the second equation of (7) we obtain s and s u ( )( u) /, where s leads to the constant y ( x) which doesn t satisfy the boundary conditions (5). Substituting s u ( )( u) / into the first equation of (7) leads to 6 ( ) (8) where u ( ) /, / u and ( / ) b. The above sixth order of algebraic equation may be changed to a cubic equation. Making use of the available analytical solution of the cubic equation (Zwillinger ), the roots of Eq. (8) are attained as ( ( (,, 5,6 ) ) ) e e.5( q i i ).5( q.5( q.5( q ) e ) e i i ).5( q.5( q and ) ) where p q p q 9. If discriminant, is positive, two roots of 6 roots are real and the others are complex. If, then the squared of all roots are real, which at least two of them are equal. If, then the squared of all roots are unequal and real. The discriminant may be rewritten as: and (9) 5 (/8)[(/ 7) (/ )[(/ 7) ] (/) (/ 7) (/ 5) ] () It is worth to mention it that there are critical values for velocity which the system should be unstable under those velocities. In this case the negative values for the frequencies of coupled system are gained. The second equation of Eqs. (7), results in s where u( )( u). In the case of, substituting and from Eqs. (9), in view of this fact that, into the relation for gives the parameters and. Therefore the solution of system of partial differential equations (6) is readily written and y( x, ) is obtained as follows y( x) [( C cos x C sin x) e i x ( C cos x C sin x) e ] e, if and i x i ()
99 R.T. FAAL and D. DERAKHSHAN / Procedia Engineering () 986 99 In the case of that or to be negative they will replace by or and then the sine and cosine functions of arguments x or x will replace with hyperbolic sine and hyperbolic cosine function with arguments x or x. It is worth to mention it that this system of equations is of fourth order and we expect for a solution with coefficients regardless of this fact that the equation s is the characteristic equation of a second order differential equation. Application of the boundary conditions (6) to the solution () leads to a system of algebraic equations as M C C C C T T where the non vanishing entries of matrix M for and, are: m, m, m i, m, m i, m, i i i i i i ( cos sin ), ( sin cos ), i i ( cos sin ), ( i sin cos ) e, m e cos, m e sin, m e cos, m e sin, m i e m i e m i e m If or to be negative then or will be replaced by ior i respectively. Finally the frequency equation is det( real( M )) or det( imag( M )). To help the memory that multiplying one of the rows of the matrix M by complex number i gives the same results for two aforementioned equations. NUMERICAL EXAMPLES AND RESULTS The analysis performed in the preceding section allows the consideration of a finite length pipe which is perfectly bonded on elastic support. Pipe and inner fluid specifications which are used for the following evaluations can be found in table (). From now on, all parameters of system in all tables are assumed to be the same as table, except for the parameters that are changing in every table or we refer to new values of them. Table : Specifications used in the examples ( Di, Do) (.5,.5) [ m], U [ m/ s], E 7[ GPa], ( p, f) (777,997)[ kg / m ], [ m], k [ N / m ] In the first example we ignore the stiffness of foundation ( k ), in this case, some of dimensionless natural frequencies of coupled system n an are tabled in table. As it mentioned before, in the earlier work (Biswas and Ahmed ) the second term of Eq. () was vanished and an analytical relation for the natural frequencies of system as n ( n u d nn ) was given where d nn [ n ( x)] dx and n is the roots of the equation cosh n cos n. Also n (x) is a mode shape as n( x) cosh nxcos nxcosh n cos nsinh nxsin nx sinh n sin n, n,,.... Comparison of this study and (Biswas and Ahmed ) shows the eliminating the second term of Eq. () ()
R.T. FAAL and D. DERAKHSHAN / Procedia Engineering () 986 99 99 can be incorrect because of missing some of the natural frequencies and also there is a difference between the others. Table. Comparison of dimensionless natural frequencies of this study and (Biswas and Ahmed ) for k. n a n 5 6 7 This study.99.577.696 7.589 56.7 78.8.8 Saroj et al. ------- 6.7995 --------- 6. --------- 9.7879 ----------- Another valuable point about the (Biswas and Ahmed ) is the determination of the stability limit. According to the relation n ( n u d nn ) the stability condition is n u d nn or u u crtical n / d nn, which for n we obtain.75 and d.5 where leads to ucrtical 6.786. For this study we set n and then we find,, and b, b, consequently we search for the fluid velocity ( u crtical ) such that the frequency equation to be satisfied at this fluid velocity. In this case for the specification of table we find u crtical.868. Therefore the velocity limit for inner fluid is U U crtical 9.79[ m / s]. The variation of ucrtical versus f A /( m p f A) as a system parameter depicted at Fig.. If the mass of inner fluid per unit length in comparison to the total mass per unit length of pipe to be small the critical velocity for the stability of system is high and reversely it would be low. The same trend can be seen in the (Biswas and Ahmed ). The effects of inner fluid velocity on the dimensionless natural frequencies of coupled system ( n an ) for velocities less thanu crtical are investigated using the two first natural frequencies of table. We may observe that the increasing the inner fluid velocity results in the decreasing of these dimensionless natural frequencies. 6 5 k= [N/m ] k=. [N/m ] u critical...6.8 β Fig. : Variation of critical inner velocity for the stability of system versus
99 R.T. FAAL and D. DERAKHSHAN / Procedia Engineering () 986 99 Table. Variation of first two frequencies withu U U 5.7.577 5.99.5 5.85.5597 5.967.79 The variation of dimensionless frequencies with the stiffness of elastic foundation and the density of inner fluid can be seen in table. As it can be seen the non dimensional natural frequencies are increased slowly when the stiffness is increased at least one order of magnitude. Another result can be found by changing the inner fluid density. As it can be observed that the dimensionless natural frequencies is reduced by growing the fluid density. Table. Variation of first two dimensionless natural frequencies with k and k f f [ kg / m ].99.577 5 (Propane).79.55..5787 997(Water).99.577.6.599 85 (Mercury).68.957 On the other hand table 5 shows that the growing of the rigidity and density of pipe enhances the dimensionless natural frequencies. The inner pipe diameter reduction (with constant thickness) and the beam length enlargement cause the weaker beam and consequently reduction in the first two dimensionless natural frequencies. This point may be realized from table 6. Table 5. Variation of first two frequencies with E and Material Aluminum E 7[ GPa], 7[ kg / m ] Table 6. Variation of first two frequencies with p E,.5 8.595 Steel E 7 [ GPa], 777 [ kg / m ]..5787 Cu E.6[ GPa], 896[ kg / m ].6.968 D and D i ( Inner diameter), t( Thickness) i [m] D i.5[ m], t.[ m]..5787..5787 D i D i.6[ m], t.[ m].9.869.95 5.6.7 [ m], t.[ m].8656.658.7.998
R.T. FAAL and D. DERAKHSHAN / Procedia Engineering () 986 99 99 CONCLUSIONS The main conclusions of this paper may be listed as follows: ) Dimensionless natural frequencies of coupled system are increased by growing stiffness of elastic support and rigidity of pipe and also by reducing inner fluid density and velocity, inner pipe diameter (with constant thickness) and beam length. ) In the stability analysis of the pipe vibration the parameter plays an important rule. References [] Ashley H and Haviland G, (95). Bending vibrations of a pipeline containing flowing fluid. Journal of Applied Mechanics. 7, 9. [] Biswas SK and Ahmed NU (). Optimal control of flow-induced vibration of pipeline. Dynamics and Control., pp. 87-. [] Gorman DG, Reese JM and Zhang YL (). Vibration of a flexible pipe conveying viscous pulsating fluid flow. Journal of Sound and Vibration., pp. 79 9. [] Lee U and Kim J (999). Dynamics of branched pipeline systems conveying internal unsteady flow. Journal of Vibration and Acoustics., pp.. [5] Lee U, Pak CH, and Hong SC (995). The dynamics of a piping system with internal unsteady flow. Journal of Sound and Vibration. 8, pp 97. [6] Lee US, Jang IN Go HA (9). Stability and dynamic analysis of oil pipelines by using spectral element method. Journal of Loss Prevention in the Process Industries,, pp.87 878. [7] Lee US and Park J (6). Spectral element modelling and analysis of a pipeline conveying internal unsteady. Journal of Fluids and Structures. pp. 7 9. [8] Zwillinger (). Standard Mathematical Tables and Formulae. CRC Press LLC.