Effect of a Viscoelastic Foundation on the Dynamic Stability of a Fluid Conveying Pipe

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1 International Journal of Applied Science and Engineering 4., : Effect of a Viscoelastic Foundation on the Dynaic Stability of a Fluid Conveying Pipe Nawras Haidar Mostafa* Mechanical Engineering Departent, College of Engineering, University of Babylon, Babel, Iraq Abstract: In this study, the natural frequency of a siply supported pipeline (hinged type) conveying one-diension incopressible steady fluid flow set on viscoelastic foundation is investigated by using finite eleent analysis and the critical fluid velocity with different paraeters such as stiffness and viscous coefficients of foundation are obtained. The foundation is siulated using the odified Winkler's odel to introduce the effect of tie dependent viscosity ter. Soe known results are confired and soe new ones obtained. Two coponents of foundation, stiffness and viscosity, seeed to act on the critical flow velocity of the pipe in contrary trend. Where, increasing the foundation stiffness tended to increase the critical flow velocity in the pipe. While, increasing foundation viscosity attepted to decrease it. At soe ranges of pipe length, the foundation viscosity effect sees to be ore extree. Increasing the fluid velocity leads to onotonic reduction in the syste daping ratio. Two paraeters, pipe length and fluid density which relate to the natural frequency of pipeline conveying fluid are studied in detail and the results indicate that the effect of Coriolis force on natural frequency is becoe ore effective by increasing pipe length and fluid density besides increasing fluid flow velocity. Keywords: Finite eleent; fluid-structural interaction; odified Winkler odel; viscoelastic foundation; siply supported pipe; stability.. Introduction Piping systes are widely utilized to convey fluids in any industrial fields, ranging fro cheical plants to biological engineering systes. Exaples include fuel pipes in engine systes, heat transfer pipes in power generation plants, refrigerators, air-conditioners, heat exchangers, cheical plants piping, hydropower systes and so forth. Piping vibration probles are therefore very iportant in industry. The instability proble of flexible pipes conveying fluid provides a paradig for the odelling and analysis of the instability echaniss of fluid-structure interaction systes. The stability and dynaic characteristics are now well understood. The dynaics are known to be sensitively dependent on flow velocity and support/boundary conditions. In general, it has been established that an initially straight pipe that conveys a fluid with a relatively low speed is stable. In other words, each disturbance applied to that pipe causes a vibration that decreases with tie. It has been also found that for fluid speed values higher than a certain value (the critical flow velocity) even a sall disturbance could * Corresponding author; e-ail: nawras98@gail.co 4 Chaoyang University of Technology, ISSN Received 9 May 3 Revised Noveber 3 Accepted Noveber 3 Int. J. Appl. Sci. Eng., 4., 59

2 Nawras Haidar Mostafa result in a syste vibration that increases with tie. In the latter circustances, therefore, the syste equilibriu state is referred as unstable. The first serious study of the dynaics of pipes conveying fluid is due to reference [] who derived the correct equations of otion and carried its analysis rearkably far, reaching adirably accurate conclusions regarding stability, in particular concerning the cantilevered syste. Major research on the regarding stability and vibration of flexible pipes conveying fluid started in the 95s in relation to the design of pipelines conveying oil. Reference [] found that the critical flow velocity of a fluid conveying pipe on Winkler foundation is higher than the critical flow velocity of that pipe without foundation. In this anner, the Winkler foundation is proved to have a stabilizing effect on the pipe. Reference [3] used the Vlasov foundation odel to describe the vibration of a pipeline containing flowing fluid and supported on an elastic foundation. The critical flow velocity of such pipe was solved analytically using interaction ethod.he concluded that elastic support would reduce that aplitude of the pipeline vibration. For pipes of a finite length, the dynaical behavior depends strongly on the type of boundary conditions at both ends [4]. References [5] and [6] have studied the dynaic stability of cantilevered pipes on foundations of constant odulus that support only a part of the pipe span. They have found that such foundations could either destabilize or stabilize the pipe depending on the position and length of the foundations. While references [7] and [8], have exained cantilevered pipes on Winkler foundations whose odulus is a certain sixth-, second- or first-order polynoial. They have concluded that all such foundations stabilize the pipe. Reference [9] studied nuerically pinned-claped and claped-pinned pipes conveying fluid. He found that to predict the dynaical behavior of the claped-pinned pipe, even 8 significant-figure accuracy was not good enough. The iaginary part of the coplex eigen frequency seeed to be negative, iplying unstable behavior for any flow velocity greater than zero. Reference [] studied the optial design of cantilevered fluid-conveying pipes. The ai of his study was to axiize the critical flow speed of the fluid by eans of additional asses, supporting springs or dapers along the length of the pipe. The optiization proble was forulated by odeling the pipe by finite eleent ethod, using Euler-Bernoulli bea eleents. The locations of the additional asses, springs and dapers and the properties of these eleents (ass, spring constant and daping constant) were chosen as design paraeters. The axiization proble for the critical fluid flow speed was solved by the sequential quadratic prograing technique. Reference [] applied the eliinated eleent-galerkin ethod to calculate the natural frequency with different boundary conditions based on typical transverse vibration odel. Then the relationship between siplified natural frequency of the pipeline and that of Euler bea was discussed. In a given boundary condition, the four coponents (ass, stiffness, length and flow velocity) which relate to the natural frequency of pipeline conveying fluid were studied in detail and the results indicate that the effect of Coriolis force on natural frequency was inappreciable. Reference [] extended the Winkler's odel to account the effect of tie dependence in the siplest case to ake it viscous. He applied the viscous odel to solve a proble of rotating wheel sets on polyer rubber sheet. Good agreeents between the proposed odel and experiental data were obtained. Reference [3] investigated the resonant vibrations of a fluid-conveying pipe, with special consideration to axial shifts in vibration phase accopanying fluid flow and various iperfections. Sall iperfections related to elastic and dissipative support conditions were specifically addressed, but the suggested approach was readily applicable to other kinds of iperfection, e.g. non-unifor stiffness or ass, non-proportional daping, weak nonlinearity, and flow pulsation. Reference [4] investigated the effects of flow velocity on the daping, stability, and frequency shift of icroscale pipes 6 Int. J. Appl. Sci. Eng., 4.,

3 Effect of a Viscoelastic Foundation on the Dynaic Stability of a Fluid Conveying Pipe containing internal fluid flow. The analysis was conducted within the context of classical continuu echanics, and the effects of structural dissipation (including thero elastic daping in hollow beas), boundary conditions, geoetry, and flow velocity on vibrations were discussed. The study showed that flow-induced daping and frequency shifts in representative single-crystal silicon structures could exceed the typical specifications for resonant icro sensors. To the best of our knowledge, other studies on dynaic stability of pipes on variable elastic foundations are not reported in the literature. It can be found in reference [5]. Fro the review of literature, it is found that the study of flow induced vibration in pipes conveying incopressible steady fluid flow ounted on viscoelastic foundation has not yet been explored so far. The ai of this is to clarify whether the critical flow velocity depends on the agnitude of the foundation stiffness, foundation daping, the pipe length, pipe thickness and fluid density for a hinged siply supported pipe conveying fluid rest on a viscoelastic foundation (odified Winkler odel).. Viscoelastic extension of Winkler foundation An analysis of the bending of beas on a viscoelastic foundation, if based on the Winkler odel, is derived fro the assuption that the foundation's reaction forces are proportional at every point to the deflection of the bea at that point. The differential equation of the elastic line is based on the assuption that a straight bea is supported along its entire length by a viscoelastic ediu and subjected to vertical forces acting in the principal plane of the syetrical cross section (see Figure ). Under these conditions, the bea will deflect, thus producing continuously distributed reaction forces in the supporting ediu. One ay ake the fundaental assuption that the intensity (p) of the reaction forces at any point is proportional to the deflection of the bea at that point. The reaction forces are assued to act vertically and in opposition to the deflection of the bea. Hence, where the deflection is directed downward (in a positive direction) the supporting ediu will be copressed. However, where the deflection is negative, tension is produced; for the purposes of this research, the supporting ediu is assued to be able to take up such tensile forces. Front View Side View Pipe Conveying Fluid W x Figure. Representation of odified Winkler foundation odel If a bea has a unifor cross section, then a unit of deflection of this bea will cause reaction in the foundation; consequently, at a point where the deflection is W(x, t), the intensity of the distributed reaction, per unit length of the bea, will be []; p( x, t ) k W dw dt () Int. J. Appl. Sci. Eng., 4., 6

4 Nawras Haidar Mostafa Where ko and μ are the stiffness and daping (viscous) coefficients of foundation per unit length respectively. The assuption p(x, t) iplies that the supporting ediu is viscoelastic. Its aterial, then, acts in accordance with Kelvin-Voigt odel. Its viscoelasticity, therefore, ay be characterized by the force which, distributed over a unit area, will cause a deflection equal to that unit. The constant values of the supporting ediu, kv and μv, are called the oduli of stiffness and viscous foundation respectively. Where, k o bk v () b v (3) The units of the oduli k v and v are in (N/3) and (N s/3) respectively. While (b) is the width of the bea in contact with the base foundation (Figure ). However, it should be reebered that k o and µ includes the effect of the width of the bea and will be nuerically equal to k v and v only if the bea is of a unit width. 3. Derivation of governing differential equation The detailed analysis of the dynaics of straight flexible pipes conveying fluid is described by references [9] and [5]. In this section, the odeling and calculation ethod based on these papers are introduced. When a pipeline rests on a viscoelastic ediu such as polyers, a odel of the viscoelastic ediu ust be included in the governing differential equation. The physical syste analyzed is shown in Figure (a). Forces and oents acting on the fluid and pipe eleents, respectively, are shown in Figure (b and c). The pipe is considered to be slender, and its lateral otions, W (x, t ), to be sall and of long wavelength copared to the diaeter. The syste consists of a unifor pipe of length ( L ), pipe ass per unit length ( ), flexural rigidity ( EI ), conveying fluid of ass per unit length ( M ), flowing axially with velocity ( U ), and ounted on viscoelastic foundation with stiffness (ko) and viscous daping (µ). The cross-sectional flow area is ( A ), inner perieter is ( S ) and the fluid pressure is ( p ). Consider then eleents x of the fluid and the pipe, as shown in Figure (b and c). The fluid eleent of Figure (b) is subjected to: (i) pressure forces, where the pressure p=p(x, t) because of frictional losses, p is easured above the abient pressure, and t is the tie; (ii) reaction forces of the pipe on the fluid noral to the fluid eleent, Fδx, and tangential to it, ( qs x ), associated with the wall-shear stress q ; (iii) gravity forces (Mgδx) in the W- direction. Balancing the forces in W-direction of the fluid eleent while keeping in ind the sall deflection approxiation, yields w F M U W pa x x t (4) The su of forces parallel to the pipes axis for constant flow velocity gives A p qs x (5) Siilarly, for pipe eleent of Figure (c) one obtains T qs x 6 Int. J. Appl. Sci. Eng., 4., (6)

5 Effect of a Viscoelastic Foundation on the Dynaic Stability of a Fluid Conveying Pipe And the forces noral to the pipe axis for sall deforation Q W W W F T k W o x t x t (7) where M 3W Q EI x x 3 (8) ( pa T ) x (9) Fro equation (5) and equation (6), the wall shear stress q is eliinated to result in The pipe end where x=l, the tension T in the pipe is zero and the fluid pressure is equal to abient pressure, thus p=t= at x=l, () pa T Cobining all the above equations yields the following governing equation EI 4W W W W W k W MU MU ( M ) o t x t x 4 x t () (a) (b) (c) Figure. (a) Siply supported pipe on viscoelastic foundation; (b) forces on fluid eleent; (c) forces and oents on pipe eleent Int. J. Appl. Sci. Eng., 4., 63

6 Nawras Haidar Mostafa The ter ( EI 4 W x 4 ) represents a force coponent acting on the pipe because of pipe bending. The ter ( k o W and W ) t represents the force coponent acting on the pipe that coes fro foundation stiffness and foundation daping respectively. The expression ( MU W ) represents x the force coponent acting on the pipe as a result of flow around a deflected pipe (curvature in pipe). The ter ( MU W ) is the inertial force associated with the Coriolis acceleration arising x t because the fluid flows with velocity U relative to the pipe. In addition, this expression is the so-called anti-syetric whirligig daping ite. Because of its effect, the fluid structural interaction odel belongs to coplex eigenvalue proble. While the expression ( ( M ) W ) t is force acting on the pipe because of inertia of the pipe and the fluid flowing through it. A rearkable feature of equation () is the total absence of fluid-frictional effects, which at first sight ight appear to be an idealization. However, within the context of the other approxiations iplicit in this linearized equation, it ay rigorously be deonstrated that fluid-frictional effects play no role in the dynaics of the syste, a fact first shown by reference [6] and [7]. 4. Finite eleent discretization equation () is a fourth-order partial differential equation in two independent variables subject to various boundary conditions. It is very difficult to get its analytical solution, while we can use finite eleent ethod to get its nuerical solution. The equation of eleent deflection for straight two diensional bea eleent could have the for [8]: W (x ) in N i ( x )q i () Where qi is the generalized coordinates ( displaceent and rotation, see Figure 3). Ni, (i=, n) are the bending shape functions and W(x) is the deforation polynoial cubic function which define the displaceents and rotations at the nodes. W q q q3 q4 x Figure 3. Bea eleent nodal displaceents shown in a positive sense Deterination of the forer, using the ethod to be found in any finite eleent texts, proceeds by first writing W(x) as an n-tered polynoial with unknown coefficients, n being the nuber of degrees of freedo in the eleent. The choice of a cubic function to describe the displaceent is not arbitrary. Clearly, with the specification of four boundary conditions, we can deterine no ore than four constants in the assued displaceent function. 64 Int. J. Appl. Sci. Eng., 4.,

7 Effect of a Viscoelastic Foundation on the Dynaic Stability of a Fluid Conveying Pipe The shape functions Ni are equal to N ( x 3 3x 3 ) 3 (3a) N ( x 3 x x ) (3b) N3 ( 3x x 3 ) 3 (3c) N4 3 ( x x ) (3d) where is the eleent length. The above shape functions, Ni, represent the conventional two-diensional bea eleents which have two degrees of freedo at each node: one lateral displaceent and one rotational. The kinetic and potential energies of the pipe eleent can be expressed by T W ( M ) t dx W EI x dx (4) q T ( M ) N T Ndxq e V (5) q T EI N T N dxq e Where each prie sign that appears above the shape function sybol, i.e. N, represents one tie derivative with respect to x-coordinate. Thus, ass ( ) and stiffness ( k ) atrices are equal to 56 4 ( M ) EI 3 k (6) (7) Over the length of elastic foundation, this adds the following ter to the total potential energy: V k ow dx (8) q T k o N T Ndxq e We recognize the stiffness ter in the above suation, k k o N T Ndx (9) Thus, the foundation stiffness atrix equals to Int. J. Appl. Sci. Eng., 4., 65

8 Nawras Haidar Mostafa 56 4 k k o The ter ( MU W x () ) has a potential energy that can be represented in ters of displaceent shape function derived for the pipe as V3 W W MU x x dx q T MU N T N dxq e () The stiffness atrix that coes fro flow around the deflected pipe is equal to 36 MU 3 k () It is iportant to clear that stiffness atrix k 3 leads to weaken the overall stiffness of the pipe syste. The expression ( W ) in equation () has a dissipation energy as t R W t dx (3) q T N T Ndxq e Leading to C N T Ndx (4) The foundation viscous atrix equals to 56 4 C (5) While the expression ( MU W ) represents the Coriolis force, which causes the fluid in the x t pipe to whip, can be represented by dissipation energy as R W W t MU x dx q e T MU N T N dxq (6) This gives the unsyetrical daping atrix MU 6 C (7) In general, the above atrices ( ass, stiffness and daping) are classified according to their 66 Int. J. Appl. Sci. Eng., 4.,

9 Effect of a Viscoelastic Foundation on the Dynaic Stability of a Fluid Conveying Pipe category. Then each category eleent atrices are assebled to represent the whole pipe length odel. The boundary conditions of the pipe ends are both siply supported (hinged type). While, the effect of viscoelastic foundation is represented as a dependent distributed load (according to equation ) along the pipe length. These overall atrices will be arranged in a proper for to get the dynaic characteristics of the structure. 5. Daped dynaic eigenvalues To analyze the dynaic eigenvalues of daped structure we should transfer the governing equation to the state-space coordinates. The standard equation of otion in the finite eleent for is ( M )q C total q k total q Where C total C C and k total k k k 3.The substitution is, [9]: (8) q (9) Resulting in the following set of equations Z dz HZ dt (3) where q Z (3) II H ( M ) ktotal ( M ) Ctotal (3) where II is a unity atrix. Therefore, we can obtain the natural frequencies (eigen values) and ode shapes (eigen vectors) by solving the atheatically well-known characteristic equation of (33) II HZ The solution of eigenvalue proble yields coplex roots. The iaginary part of these roots represents the natural frequencies of daped syste. The real part indicates the rate of decay of the free vibration. 6. Logarithic decreent A convenient way of deterining the daping in a syste is to easure the rate of decay of oscillation (the real part eigenvalue). The logarithic decreent,, is the natural logarith of the ratio of any two successive aplitudes in the sae direction, where Y, and Y, are successive aplitudes, where [] Y ln Y (34) As entioned before, the rate of decay is obtained fro solving the characteristic equation (equation 33). So that the left hand ter in equation (34) is already known and we should only Int. J. Appl. Sci. Eng., 4., 67

10 Nawras Haidar Mostafa find the daping ratio ( ) fro the above equation. 7. Result and discussion Figure 4(a) shows the relation between fluid velocity and the pipe frequency with various foundation stiffness of siply supported pipe ounted on viscoelastic foundation. As the foundation stiffness is increase, the natural frequency of the pipe is also increase. This behavior is taken place since the global stiffness of the syste is increased. Furtherore, the relation between foundation stiffness and pipe frequency sees to be linear at constant fluid velocities as depicted in Figure 4(b). 6 Pipe length is ( ) Fluid density is ( kg/3) Pipe density is (8 kg/3) Pipe thickness is (. ) Outer diaeter of the pipe is (. ) Elastic odulus of pipe is (7 GPa) µv, is equal to ( N.s/3) kv(kn/3) 3 Natural Frequency (rad/sec) Fluid Velocity (/s) (a) 6 Fluid Velocity (/s) U= U=4 Natural Frequency (rad/s) 5 U= Foundation Stiffness (kn/3) (b) Figure 4. Effect of foundation stiffness on the natural frequency of the pipe at different fluid velocities 68 Int. J. Appl. Sci. Eng., 4.,

11 Effect of a Viscoelastic Foundation on the Dynaic Stability of a Fluid Conveying Pipe Figure 5 shows the effect of viscous daping coefficient of the foundation on the first natural frequency of the pipe syste with different fluid velocities. As it is well known, the daped natural frequencies of daped syste are saller than it is for undaped one. Thus, an increase in the foundation viscous coefficient leads to reduce the dynaic properties of the pipe syste. This behavior is not always true for daped syste. Where the daped natural frequency of the lowest ode ay be higher than the corresponding undaped frequency depending on the choice of daping atrix and the ode separation []. 5 Pipe length is ( ), Fluid density is ( kg/3) Pipe density is (8 kg/3) Pipe thickness is (. ) Outer diaeter of the pipe is (. ) Elastic odulus of pipe is (7 GPa) Foundation stiffness coefficient ( kn /3) Natural Frequency (rad/sec) v (kn.s/3) Fluid Velocity (/s) Figure 5. Effect of foundation viscous daping on the natural frequency of the pipe at different fluid velocities Figure 6 presents the relation between pipe length and critical flow velocities for different foundation properties. As well known for a pipe without foundation, increasing the pipe length leads to reduce the pipe stiffness and raise its ass thus decreasing the critical flow velocity. This behavior is differing for a pipe with viscoelastic foundation. Where there is a reduction in the critical flow velocities with increasing the pipe length then at soe pipe length the critical velocity start to rise, then after reducing again and continue with copacted values. This convexity in the curve is ainly caused by the foundation stiffness and this behavior is copletely agree with the study that done by [3]. Moreover, when increasing foundation daping, the critical velocities exhibit ore reduction in their values than it for sall daping does. This behavior is caused by increasing the overall daping of the syste that leads to decrease its daped natural frequency. Thus, we can say that, in viscoelastic foundation, daping induces destabilization effect, while foundation stiffness leads to stabilize the pipe. It has thus found that the criterion for global instability as the length is increased becoes closely related to the local properties of the waves in the pipe []. It is iportant to record that at soe ranges of pipe length, the foundation viscosity effect sees extree and obvious. Int. J. Appl. Sci. Eng., 4., 69

12 Nawras Haidar Mostafa Foundation Properties kv=5 kn/3, v= kv=5 kn/3, v=5 N.s/3 5 kv=5 kn/3, v= N.s/3 kv=3 kn/3, v= N.s/3 Critical Velocity (/s) kv=3 kn/3, v=5 N.s/3 kv=3 kn/3, v= N.s/3 kv= kn/3, v= N.s/3 kv= kn/3, v=5 N.s/3 kv= kn/3, v= N.s/3 7 Pipe density is (8 kg/3) Fluid density is ( kg/3) Outer diaeter of the pipe is (. ) Pipe thickness is (. ) Elastic odulus of pipe is (7 GPa) Pipe Length () Figure 6. Effect of pipe length on the critical fluid velocity for different ounting conditions Figure (7) shows the effect of fluid velocity on the syste's daping ratio for two different foundation viscosities. The daping ratio will decrease onotonically with increasing fluid velocity. This event is ainly caused by decreasing the rate decay of pipe vibration..44 Pipe density is (8 kg/3) Fluid density is ( kg/3) Outer diaeter of the pipe is (. ) Pipe thickness is (. ) Elastic odulus of pipe is (7 GPa) Pipe length is ( ).4 Daping Ratio ( ).4.38 Pipe length = kv=3 kn/3, v=5 N s/3.36 kv=3 kn/3, v= N s/3 Critical Velocity Fluid Velocity (/sec) Figure 7. Effect of fluid velocity on the syste's daping ratio Figures 8 (a) and (b) show the percentage errors in the predicted natural frequencies of the pipe syste with neglecting the Coriolis coponent for different pipe lengths and fluid densities 7 Int. J. Appl. Sci. Eng., 4.,

13 Effect of a Viscoelastic Foundation on the Dynaic Stability of a Fluid Conveying Pipe respectively. Fro these figures, a pipe with larger length and higher fluid density has the biggest frequency percentage error. Furtherore, increasing the flow fluid velocity leads to increase the percentage error. Pipe density is (8 kg/3) Fluid density is ( kg/3) Outer diaeter of the pipe is (. ) Elastic odulus of pipe is (7 GPa) Foundation stiffness coefficient, kv, is ( kn /3) Foundation daping coefficient, µv, is ( N.s/3) Pipe Length L =.7 L =.3 Percentage Error L= Fluid Velocity (/s) (a) Pipe density is (8 kg/3) Outer diaeter of the pipe is (. ) Elastic odulus of pipe is (7 GPa) Foundation stiffness coefficient, kv, is ( kn /3) Foundation daping coefficient, µv, is ( N.s/3) Fluid Density (kg/3) 68 6 Percentage Error Fluid Velocity (/s) (b) Figure 8. Effect of whether considering coriolis force in (a)different pipe lengths; (b)different fluid densities 8. Conclusions The effects of a viscoelastic foundation on the stability of a fluid conveying pipe were analyzed by using the extension of Winkler foundation odel. The proble was analyzed nuerically using finite eleent ethod. Soe interesting conclusions have been drawn, as Int. J. Appl. Sci. Eng., 4., 7

14 Nawras Haidar Mostafa follows: a) The foundation stiffness leads to increase the fluid critical velocity, while foundation daping decreases it. b) There are ranged values of pipe length for each foundation properties that sees to be ore affected by the foundation characteristics. c) Coriolis coponent still plays a ajor role in the dynaic behavior of pipe especially with larger length and heavier fluid. d) The daping ratio of the syste is decreased onotonically with increasing the fluid velocity. 9. Noenclatures 7 Sybol A b C C e E f (x, t) F g H I II ko Definition Cross-sectional flow area Width of the bea in contact with the base foundation Foundation viscous atrix Units - Daping atrix caused by Coriolis force - Eleent Modulus of elasticity of pipe Intensity of reaction force of foundation Reaction force inside the pipe Acceleration constant Characteristic atrix Pipe second oent of area Unity atrix Foundation stiffness coefficient per unit length N/ N/ N /s 4 N/ kv k k k3 Foundation stiffness coefficient per unit area N/3 Stiffness atrix of pipe - Foundation stiffness atrix - Stiffness atrix coes fro flow inside deflected pipe - L M M Ni p Q q q q q Length of the pipe Eleent length of pipe Fluid ass per unit length Pipe ass per unit length Pipe ass atrix Bending oent Shape function Pressure inside the pipe Shear force Wall shear stress Lateral displaceent of pipe Lateral velocity of pipe Lateral acceleration of pipe kg/ kg/ N N/ /s /s N N/ /s /s Int. J. Appl. Sci. Eng., 4.,

15 Effect of a Viscoelastic Foundation on the Dynaic Stability of a Fluid Conveying Pipe S T t U x, W v Pipe inner perieter Tension force in the pipe Tie Fluid velocity relative to the pipe Cartesian axes Foundation daping coefficient per unit length Foundation stiffness coefficient per unit area N s /s N s/ N s/3 Eigen values Daping ratio - Acknowledgeents The authors wish to acknowledge fruitful discussions with Professor Michael P. Païdoussis (Departent of Mechanical Engineering, McGill University). References [ ] Bourrieres, F. J Sur Un Phenoene D oscillation Auto-Entretenue En Mecanique Des Fluids Reels. Publications Scientifiques et Techniques du Ministere de L Air, (47). French. [ ] Lottati, I. and Kornecki, A. 986.The effect of an elastic foundation and of dissipative forces on the stability of fluid conveying pipes. Journal of Sound and Vibration, 9(): [ 3] Chen, H. 99. Vibration of a Pipeline Containing Fluid Flow with Elastic Support. M. Sc. Thesis, Ohio University. [ 4] Lee, S. Y. and Mote, C. D A generalized treatent of the energetics of translating continua, part II: beas and fluid conveying pipes. Journal of Sound and Vibration, 4(5): [ 5] Elishakoff, I. and Ipolonia, N.. Does a partial elastic foundation increase the flutter velocity of a pipe conveying fluid? Journal of Applied Mechanics-Transactions of the ASME, 68(): 6-. [ 6] Djondjorov, P.. Dynaic stability of pipes partly resting on winkler foundation. Journal of Theoretical and Applied Mechanics, 3(3): -. [ 7] Djondjorov, P., Vassilev, V., and Dzhupanov, V.. Dynaic stability of fluid conveying cantilevered pipes on elastic foundations. Journal of Sound and Vibration, 47(3): [ 8] Djondjorov, P.. On the critical velocities of pipes on variable elastic foundation. Journal of Theoretical and Applied Mechanics, 3(4): [ 9] Païdoussis, M. P. 4. Fluid-Structure Interactions. Vol.. Acadeic Press: London. [] Luijärvi, J. 6. Optiization of Critical Flow Velocity in Cantilevered Fluid Conveying Pipes, with a Subsequent Non-Linear Analysis. Acadeic Dissertation to be presented with the assent of the Faculty of Technology, FI-94 University of Oulu, Finland. [] Huang, Y., Liu, Y., Li, B., Li, Y., and Yue, Z.. Natural frequency analysis of fluid conveying pipeline with different boundary conditions. Nuclear Engineering and Design, 4(3): [] Mahrenholtz, O. H.. Bea on viscoelastic foundation: an extension of Winkler s Int. J. Appl. Sci. Eng., 4., 73

16 Nawras Haidar Mostafa [3] [4] [5] [6] [7] [8] [9] [] [] [] 74 odel. Archive of Applied Mechanics, 8(): 93-. Thosen, J. and Dahl, J.. Analytical predictions for vibration phase shifts along fluid-conveying pipes due to Coriolis forces and iperfections. Journal of Sound and Vibration, 39(5): Rinaldi, S., Prabhakar, S., Vengallatore, S., and Michael, P.. Dynaics of icroscale pipes containing internal fluid flow: Daping, frequency shift, and stability. Journal of Sound and Vibration, 39, 8: Païdoussi, M. P Fluid-Structure Interactions, Slender Structures and Axial Flow. Volue One, Acadeic Press: California, USA. Benjain, T. B. 96. Dynaics of a syste of articulated pipes conveying fluid: theory. (a). Proceedings of the Royal Society (London) A, 6, 37: Benjain, T. B. 96. Dynaics of a syste of articulated pipes conveying fluid. (b). Proceedings of the Royal Society (London) A, 6, 37: Rao, S. S. 4. The Finite Eleent Method in Engineering. 4th ed. Elsevier Science & Technology Books. Caughey, T. K. and O'Kelly, E. J. 96. Effect of daping on the natural frequencies of linear dynaic systes. Journal of the Acoustical Society of Aerica, 33, : Beards, C. F Structural Vibration: Analysis and Daping. st ed. Butterworth-Heineann: An iprint of Elsevier Science. Krodkiewski, J. M., 8. Mechanical Vibrations. st ed. The University of Melbourne, Departent of Mechanical and Manufacturing Engineering: Melbourne. Doared, O. and De Langre, E.. Local and global instability of fluid-conveying pipes on elastic foundations. Journal of Fluids and Structures, 6(): -4. Int. J. Appl. Sci. Eng., 4.,