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The dynamic stability of a pipe conveying a pulsatile flow
fnr.J.
Engng Sci.,
1973, Vol. 1 Lpp. 1013-1024.
THE DYNAMIC
Pexpmon Press, PrintedinGreat Britain
STABILITY OF A PIPE CONVEYING PULSATILE FLOW JERRY
A
H. GINSBERG
Purdue University, School of Mechanical Engineering, West Lafayette, Indiana 47907, U.S.A. West Lafayette, Indiana 47907, U.S.A. Abstract-This investigation concern the small displacements of a pipe conveying a pressurized flow whose velocity possesses a harmonic fluctuation about a mean value. A new derivation of the fluid forces is used to obtain the partial differential equation of motion. General equations applicable to any set of boundary conditions are specialized for the case of a simply supported pipe and the Galerkin method is utilized to find solutions. An analysis of the case of steady flow shows that the pipe exhibits the divergence type of instability which is predictabie by static structural theory. In the presence of pulsatileflow the pipe has regions of dynamic instability whose extent increases with increased amounts of fluctuation, The results have great similarity to those for beams carrying pulsating end forces.
INTRODUCTION THE WBRATION
and stability of pipes conveying flowing fluid fiows has been the subject of a long series of investigations, see [l-9] for example. These studies have considered the effects of fluid velocity, pressure, viscosity, and compressibility upon both infinitely long pipes and pipes with various types of boundary conditions. However, in each case the equations of motion were solved for the case of a fluid flowing at a constant speed. This study was motivated by the fact that it is not likely that a pump will yield a truly constant velocity. In fact for some pumps the velocity fluctuation is quite appreciable. The problem investigated herein concerns the parametric excitation of the transverse displacement of a pipe due to a pulsating flow. This results in the possib~ity of an oscillatory instability at mean flow velocities below the critical one for steady flow. In general, transverse motion of the pipe will affect the character of the flow, but it is assumed that this effect is negligible. The fluid is considered to be incompressible and inviscid with a uniform velocity profile. The pipe has a constant cross-section which is completely filled by the fluid. There is a pump at the end s = 0, and the end s = L is connected to a chamber which is maintained at a constant pressure pO. General equations of motion for small transverse displacements are derived for any set of end conditions, and solutions are obtained for the case of simple supports. Although the equation for the transverse force exerted by the flowing fluid upon the pipe has been derived elsewhere, a new derivation is presented here. It is believed that this derivation will prove to be extremely useful in an analysis of the effects of large transverse displacements upon the predicted regions of dynamic instability, EQUATIONS
OF MOTION
Let the pipe be in a deformed position. Consider a segment of fluid of length ds at a d$$tnn_ces from the pump. Let XYZ be a space fixed reference frame with unit vectors I, f, K and let xyz be a reference frame attached to the pipe with unit vectors zx 2, as shown in Fig. la. The acceleration of the fluid is given by 1013
IJEsVd
11 No9-F
1014
JERRY
H. GINSBERG
If the diameter of the pipe is sufficiently small compared to the length, then Go will be negligible. Thus, letting u and w be the Lagrangian components of the displacement of point 0, the remaining terms in equation (1) are
Fig. la. Coordinate systems and geometry for a deformed section of pipe and fluid.
Fig. 1b. Free body diagram of an elemental slug of flowing fluid.
The dynamic stability of a pipe conveying a pulsatile flow
When jw/L( Q 1, cos r#~= 1, sin 4 = 0, ?-
1015
i, c== f?, Z = 0, 4 = awl&, and l/R =
a%/as? Now consider
a free body diagram of the slug of fluid, Fig. lb. Equilibrium in the z direction requires that the transverse force per unit length _/be given by (3) where p is the density of the fluid and A is the internal cross-sectional area of the pipe. Substituting equations (1) and (2) and applying the conditions for small displacements then yields (4) The velocity and pressure of the fluid are related to the output of the pump by the laws of fluid dynamics. The pump produces a harmonic pulsation in the flow velocity, and the open end of the pipe is at a gauge pressure p. z+((s=O,r)
= vo+v,sinot (9
p(s=L,t)
=po.
The continuity condition for the incompressible
inviscid fluid requires that
Q(S, t) = ~(0, t) = vo+ u1sin ot
(6)
while equilibrium tangent to the axis of the pipe requires that
ap p!% -=as
at.
(7)
The solution of equation (7) which satisfies equations (5) and (6) is p(s. t) =po+pov,(L-.s)
cosot.
(8)
Substitution of equations (6) and (8) into equation (4) then yields an expression for the transverse force. It is assumed that the material of the pipe obeys a stress-strain relationship of the Kelvin type, i.e.,
In the case of small transverse displacement pipe may be obtained from [ 1 I]
the equation for dynamic equilibrium of the
(10) where I is the second moment of area and m is the mass per unit length of the pipe.
1016
JERRY
For convenience
H. GINSBERG
define the following non-dimensional
e=$
7 =
p=e,
e=d,
f&t,
parameters:
y=E.L, 1
(11)
+y,
a=PA
pA+m
where Oj is the jth natural frequency of the pipe when the fluid is at rest and unpressurized. In the case of a simply supported pipe
1* l/2
wj
=
j2f12
(pA?m)L4
After performing the necessary substitutions
‘+ (pA
(12)
the equation of motion becomes
+:)LaO;
(wi”+GwiL’)+[cQZ(1+2ysin&+y2sin2&) +~+o~~e(l-~)cos~~]w”+2o~(l+ysin8~)~’=0.
(13)
A dot and a prime indicate differentiation with respect to T and 5, respectively. To determine the response the displacement w is expanded in an infinite series of the normal modes of free vibration of the pipe when the fluid is at rest and unpressurized w =
2 ~j(()uj(T);
c#q- (pA+Ey4+#Jj = 0.
(14)
j=l
For the case of a simply supported pipe +j({) = sin j9r[.
(15)
Now substitute equation (14) into equation (13), multiply the result by c#Q(~)(i is any positive integer), integrate over 0 S 4 S 1, and use the orthogonality condition for the modes. The result is &+~(&&+a,)+20$(l+ysin&) 4
i
pijcij
j=l
+[a/3z(l+2ysin&+y2sin2&)+~l
2
qijaj
i=l
+
apye
cos
87 i j=l
where
r{jUj
= 0;
i= 1,2,.
. .
(16)
1017
The dynamic stability of a pipe conveying a pulsatile flow
(17)
It is obviously not possible to work with equation (16) unless i andjare truncated at some finite value, resulting in a solution by the Galerkin method. For the simply supported pipe, the Euler buckling modes are identical to the modes of free vibration. Thus, a two term truncation will result in stability criterion which are reasonably accurate for instabilities associated with the first mode, and less accurate for instabilities associated with the second mode [ 123.In this case equations (16) and (17) give ~,+6h,-~~p(l+~sin8~)d,+[~~-~201p2y(2sin6)7+ysin2~7) +27rz+
l)arprecos &-la, -?&pye(cos
&)a, = 0
The values a2, and a2, correspond to the first two natural frequencies of a pipe carrying a steady flow obtained from an analysis which neglects the effect of Coriolis acceleration.
sz,= [1 -+(ap”+&&)]“z (19) 112
a,=4
1-$ap’+&&)
.
I
I
It should be noted that for the simply supported pipe pfj = 0 if i = j, so that the effect of the Coriolis acceleration of the fluid would be unaccounted for if only one term in equation (14) was retained. STEADY
FLOW
When the velocity of the fluid with respect to the pipe is constant, y vanishes. The equations of motion (18) then reduce to a pair of coupled ordinary differential equations with constant coefficients. Their solution is q(7)
= i
aijeV;
.i=i
i=2,2
(201
where the values A,are the roots of the characteristic equation h4+ 17SA3+(n:+@-t-
16i32-t-~~2j32)h2+ (16~::+~,2)SA+Q~~n,2= 0.
(2 1)
1018
JERRY
H. GINSBERG
The solutions (20) are stable if Re (Xj) c 0 (j = 1 - 4). Because the coefficients of equation (21) are real, the roots always occur as two pairs of complex conjugates. First consider the situation when damping is absent, 6 = 0, in which case equation (21) is biquadratic. In order for the response to be stable it must be true that both values of A2be negative. This criterion is potentially satisfied in two regions. The first region corresponds to the case 0; > 0, which is satisfied if p < p* where p* =
[;($-1*)]1’2.
(22)
Any value of p exceeding that of equation (22) corresponds to the static instability of divergence, previously investigated in reference [6]. If p > l/,rr* the response is always statically unstable. It is also possible for A*to be negative for values of /3 slightly greater than that which gives fli = 0,
P>
[-$qp.
(23)
This situation occurs whenever the coefficient of A* in equation (21) is positive for fii = 0. Substitution of equation (19) reveals that this small region of dynamic stability is obtained whenever 27 k
4a
(24)
A typical case is illustrated in Fig. 2. The listed values of the parameters approximately represent a 15 ft, 6 in. diameter steel pipe with O-05 in. wall thickness when p. = 100 p.s.i. Divergence occurs at /3 = 0.338 and the small region of dynamic stability occurs for 0.703 6 p G 0.741. It should be noted that for two reasons this latter region is of academic interest only. First, as noted previously it is statically unstable, and second, it disappears when even a small amount of damping is present in the system. When 6 # 0 the two pairs of complex conjugate roots of equation (21) always have non-zero real parts, except when CIf = 0 (p = /?*> or fit = 0. Stability requires that each root satisfy Re (A) < 0, which is found only for p < p*. The topology of the graphs of lZm(A) ) vs /3 and Re (A) vs j3 is essentially the same as when 6 = 0, except for values of /3 greater than that which gives fi22= 0. In this region the value of Re (A) always bifurcates upward from the point Re(A) = 0, so that one branch of Re(A) always corresponds to an unstable solution. In Fig. 2 the values of (Im (A) ( and Re(A) are compared for the case 6 = 0 and for a typical small value 6 = O+IO5. PULSATING
FLOW
When the flow is unsteady (y # 0) general solutions of equation (18) are not available. However, here we are concerned only with the question of whether the undeformed position is stable in the sense of Liapunov, i.e. is ai bounded for all values of 7. Using Floquet’s theory [13, 141 it is easy to show that since the coefficients
The dynamic stability of a pipe conveying a pulsatile flow
1019
-0,
-0,
4-
-0
RdXkJ
3-
\ 1; e4 E L:
--(
\
--___-8-0 z-
-6=om5 --
\
S=O and 6=0.005
I
\
--/
roc “4
I\
Im(X)
\
\
PC non-dimensional
im(X)\j \ I \
fluid velocity)
Fig. 2. Roots of the frequency equation for steady flow when o! = 0.8 and p = 0.01.
of equation (18) are periodic in T = 2?~/0,transition from stability to instability will be marked by the existence of solutions which have a period of T or 2T. Thus, the values of the parameters which define the stability boundaries are obtained by seeking conditions under which such periodic solutions exist. Following the method described in [12] solutions whose period is 2T may be constructed from the Fourier series CZt(7)= i
j=l
[bij COST+
Cij sin’+] ;
i=
1,2
(2.5)
while solutions whose period is T may be constructed from the Fourier series a, (7) = 4dfo-I-2 [du cosj&+fij
j=1
sinj&J;
i= 1,2.
(26)
In the limit as p + 0 (slow fluid velocity) regions of instability are found to exist in the vicinityofo=2wIMando=2&j, (j--1,2,.. .). The regions corresponding toj = 1 are called the principal instability because they have the widest extent and also because they are least affected by damping. This study will be confined to investigating the p~ncipal region of instability and also the first region of secondary instability (.j = 2).
1020
JERRY
H. GINSBERG
If the value of p is sufficiently small compared to p* (i.e. if the steady component of fluid velocity is not too close to the divergence condition) reasonable accuracy in the determination of the principal region will be obtained by truncating equation (25) atj= 1 ~~(7) = 6,, cos$+c,,
sin:;
i= 1,2.
(27)
This expression is substituted into the equations of motion (18) and the method of harmonic balance is used to obtain simultaneous homogeneous equations for the coefficients bil and cil. For a nontrivial solution the determinant of their coefficients must vanish jkij[ = 0
(28)
where 62
1 1)&O--~p*@y*
1
k,, = fl,z-,$2.rr2+ k,, = tti0 -
rr*c@*y
k,, = -k,,
= -6aPy9
k,, = -k,,
= -k,,
= k,, = -$a@’
k,, = -883 - IT*c@“~
(29) k,, = -k,,
= -F&0
k,, = &&$r*+
l)a~yO-27r2a/32y2
k,,= 888 - 4r2a/3*y k43 = -860 - 41~*01~~y
k,, = a;-;+$(87r*+
Y +
1)a/3yO-27r2~~2y2.
For the case without damping, evaluation of equations (28) and (29) in the limit 0 yields (k,, k,, - k,“, ) (k,, k,, - kf4 ) = 0. Both factors give the same result.
(g4-(ng+n:+~a*~*)(~)2+n~~~ =0.
(30)
Comparing the above with equation (2 l), it is seen that (8/2) * = -A*. Therefore, twice the frequencies of free vibration for the steady flow case when /3 < /3* form the back-
The dynamic stability of a pipe conveying a pulsatiie
1021
flow
bone curves for the regions of the principal ins~b~ity. It will be seen that y controlis the width of these regions, The secondary regions of instability may be studied in a similar manner. Again for sufficiently small p, equation (26) may be truncated atj = I U,(7) = frdio+ d,, cos & +.f& sin 67.
(31)
The above is substituted into equations (18) and six homogeneous equations for the coefficients are obtained by the method of harmonic balance. The boundaries of the regions of secondary instability are then obtained by solving for the values of 8 which cause the dete~inant of the system of equations to vanish. It is found that when damping is not present the limit y + 0 yields e4-- (n~+n,z+yay32)e2+Qg2;
= 0
(32)
so that the backbone curves for the regions of secondary instability are the same as the frequencies of free vib~tion for steady Bow when j3 < ,@*.Extending this result it may be shown that 2 [1m(h)l/jform the backbone curves for thejth regions of instability. In general the principal regions of instability are of greatest concern, They have the greatest width and are thus most likely to occur, as is shown in Fig. 3. A study of the
______
s 86
850
--
2
23
a=002
6=Oond
6=0.02
6 2 5 s0 4 c 0
______
----^-Lc---~-
-__-<====----e_ - _ _ _ _ _==-_-_-=<
.-s F
E
0
.&2 t? 8 ___c_L_--_____________
-~ 0
1
-vm /
005 p
---==-==X~ I O-15
1
51
(non-dimensional
fluid
/ 0.2
J 0.25
velocity)
Fig. 3. The effect of damping on the stability boundaries for pulsatile flow when (Y= 0.8, /.L= 0.01 and y = 0.4.
1022
JERRY
H. GINSBERG
effect of damping shows that when 6 = 0 there are values of 0 representing unstable responses for every value of p, whereas when damping is present p must attain a critical value in order for any instability to occur. This critical value increases as 6 increases. Furthermore, damping in general has greater effect upon regions of instability associated with the higher natural frequencies and also upon higher order regions of instability. In fact, for the example of Fig. 3, the secondary region associated with the second natural frequency has disappeared. The effect of different amounts of pulsation is illustrated in Fig. 4, wherein it is seen that a 100 per cent fluctuation (y = 1) results in relatively wide regions of instability and a 10 per cent fluctuation results in narrow regions. For brevity diagrams illustrating the effects of each parameter are not presented. Noting that the backbone curves for each region are described by equations (30) and (32), it was found that the main effect of increasing the pressure (increasing CL)was to decrease the scale of the ordinate 0 and absisca /3; lowering the density of the fluid (decreasing (u) increased the scale of the absisca /3 only. DISCUSSION
The investigation of a pipe conveying a pulsating fluid flow has shown that for values of mean flow velocity that are small compared to the one causing static instability
-L 025 p (non-dimensional
fluid
velocityl
Fig. 4. The effect of the amount of pulsation in the flow velocity on the stability boundaries when a = 0.8, w = 0.01, and 6= 0.02.
The dynamic stability of a pipe conveying a pulsatile flow
1023
(p < p*), there is little chance of an ins~bility occurring for two reasons. First, because all systems have some damping, regions of instability exist only if j3 exceeds a certain critical value. Second, even if /3 exceeds this value, the range of frequencies which correspond to unstable situations is extremely limited for the usual amounts of velocity fluctuation. As the value of fl is increased the range of unstable frequencies widens, so that dynamic instability becomes more possible. However, it should be remembered that for higher values of j3 additional terms in the time series for a, (7) and a2 (T), equations (25) and (26), must be retained. Thus, in this case Figs. 3 and 4 must be used with care. Also, the regions of instability associated with the second natural frequency are less accurate than those associated with the fundamental frequency, because the higher modes were not retained in the analysis. A comparison of the results for the pipe to those for the problem of a beam carrying a pulsating follower force1121 deserves consideration. The forces exerted upon both structures are nonconservative. In the absence of fluctuation in the value of the end force or flow velocity, instability for structures that are simply supported may be predicted by a static analysis, whereas for other types of supports instability can be correctly predicted only by a dynamic analysis. When pulsations are present, the nonconservative aspects of the forces merely modify the extent of the regions of instability and do not cause any new phenomena. Furthermore, the diagrams of the regions of instability for the two problems have very similar patterns. Interesting topics for continuing study are the effects of fluid viscosity and turbulence. Viscous effects were shown to be unimportant for steady flow [4,7], but it is not clear that this will be true for pulsatile flow. This study can be considered to describe the qualitative effects of turbulence, but a more accurate study of this matter would require solution of the equations of motion when the time dependent coefficients are stochastic, rather than harmonic, functions. REFERENCES 111 H. ASHLEY and G. HAVILAND, 1. uppl. Mech. 17,229 (1950). 123 G. W. HOUSNER, J, uppI. Me&. 19,205 (1952). [3] F. I. N. NIORDSON, Trans. R. Inst. Tech. 73(1953). [4] T. B. BENJAMIN, Proc. R. Sot. Ser. A, 261,457 and 487 (1961). 151 V. W. ROTH, Ing.-Archiu33,236 (1964). f61 H. L. DODDS, Jr. and H. L. RUNYAN, Technical Note D-2870, National Aeronautics and Administration (1965). [7] R. W. GREGORY and M. P. PAIDOUSSIS, Proc. R. Sot. Ser. A, 293,512 and 528 (1966). [SJ S. NEMAT-NASSER, S. N. PRASAD, and G. HERRMANN, AIAAJ. 4,1276 (1966). [9] S. NAGULESWA~N and C. J. H. WILLIAMS, J. me&. Engng Sci. 10,228 (1968). 1101 R. A. STEIN and M. W. TOBRINER, 1. uppE. Mech. 37,906 (1970). [l I] W. NOWACKI, Dynamics ofElastic Systems. Chapman & Hall (1963). [12] V. V. BOLOTIN, Dynamic Stability of Elastic Systems. Holden-Day (1964). ]13] R. A. STRUBLE, ~oali~eur Partial Dz~ere~r~afEquations. McGraw-Hill (1962). ii41 L. MEIROVTICH, Methods ofAnalytical Dynamics. McGraw-Hill (1970).
Space
(Received 24 October 1972) R&unt!-Cette recherche conceme les petits deplacements d’un tube conduisant un ecoulement sous pression dont la vitesse presente une fluctuation harmonique autour d’une valeur moyenne. Une nouvelle derivation des forces du liquide est utilisee pour obtenir l’iquation aux dilferentielles partielles du mouvement. Des equations g&&ales applicables a tout ensemble de conditions aux limites sont adapt&es au cas d’un tube simplement support& et h methode de Galerkin est utilisee pour trouver des solutions. Une
1024
JERRY
H. GINSBERG
anaiyse du cas de I’icoulement stable montre que le tube pr&ente le type d’instabilite de la divergence, que l’on peut prevoir par une theorie structurelle claasique. En presence d’un ecoulement pulsatoire, le tube a des regions d’instabilite dynamique dont l’itendue s’accroit avec une augmentation de la quantite des fluctuations. Les resultats ont une grande similitude avec ceux des poutres supportant en leur extritmite des forces pulsatoirea. Zusammenfassung- Diese Untersuchung betrifft die kleinen Verdr%gungen eines Rohres, das einen unter Druck stehenden Strom fdrdert, dessen Geschwindigkeit eine harmonische Schwankung urn einen Mittelwert aufweist. Es wird eine neue Ableitung der Fl~ssigkeitskr~te verwendet, urn die partielle Differentialgleichung der Bewegung zu erhalten. Allgemeine, auf jeden Satz von Grenzbedingungen anwendbare Gleichungen werden auf den Fall eines einfachen unterstiitzten Rohres spezialisiert und die Galerkin’sche Methode wird zum Auffinden von Lijsungen verwendet. Eine Analyse des Falles stetigen Stromes zeigt. dass das Rohr den Divergenztyp von ~nstabil~t~t aufweist, der durch statische St~ktu~heo~e vorhersagbar ist. in Gegenwart von Pulsierstrom hat das Rohr zonen dynamischer Instabilitat, deren Ausdehnung sich mit vergriisserten Schwankungsmengen vergrossert. Die Resultate sind den Resultaten fiir pulsierende Endkr%fte tragende Balken sehr Ihnlich. Sonmario - Quest0 studio verte sui piccofi spostamenti di un t&o the trasporta un flusso pressurizzato la cui velocita ha una fluttauzione armonica intomo a un valore medio. Una nuova derivazione delle forze fluide viene impiegata per ottenere I’equazione differenziale parziale del moto. Si specializzano equazioni generali applicabili a qualsiasi gruppo di condizioni limite per il case di un tubo a semplice supporto e si usa il metodo di Galerkin per trovare le soluzioni. Un’analisi de1 case di flusso unifo~e dimostra the il tubo mostra il tipo di divergenza d’instabilita the & prevedibile mediante la teoria strutturale statica. In presenza del flusso pulsatile il tubo ha regioni d’instabilit8 dinamica il cui valore aumenta con I’aumento nella fluttuazione. I risultati si avvicinano molto a quelli delle travi the portano forze assiali pulsante. t%CTpWTGoo6maer 0 He6OnbUILiX CMeu&HKRX Tpy6bI B CnyYW, KOrAa CKOpOCTb lIOTOKa.npH ROBbIIlIeHHOM AitBJleHBH BMeeT rapMOHHYeCKOe Kone6ame OTHOCHTeJIbNO CpeAHerO 3HaYeHWI. Ha OCHOBe HOBOPO BbIBOAa H(HAKOCTHblX CUII llOJlyYaeTC8lAH~&WfWiaJIbH~ ypaBHeHKe ABEKeHMJl B YaCTHblX npOH3BOAHbIX. &PI uOMep~U~HH0~ Tpy6bI OCO6eHHO EOIIyYZUOTCSlo6mne ypaBHeH&iR lIpUMeH%iMb!e K npO~3BOAbHO~ cucfetde rpaH~YHb~X ycnomi%, peruembie npE rioMouw MeTaDa RmepKmia. @is aHami3a cny%w CTauwoHapHoro I'lOTOKarpy6a npOflBlI~eT yCTOi%lBOCT% JWiBepreHTHOrO Tllna, YTO B03MO?KHO noToKa npe.IWKa3aTb, liCIIOJIb3ySI CTaTHYeCKyH, CTpyKTypHyHJ TeOpUIO. IIplr HamiYmi nynbcEcpymuer0 Tpy6a &iMeeT o6nacTu AEiHaMllYeCKOi%)'CTOk'ftiBOCTIi, IIpOTSDKeHEieKOTOpbIX yBeJ&iYaeTCffC yBeJEiYeHHeM Kosre6~uii.rrpu3TOMahAeeTCA3HaMeSIITenbHOe nono6we Me~Ay3T~M~~3YAbTaTaM~~TeM~ AJIRlQ”F3G HeCy3IJEiX IIyJIbCSipyIoWue KOHeYHbIeCIIJIbI.
Engng Sci.,
1973, Vol. 1 Lpp. 1013-1024.
THE DYNAMIC
Pexpmon Press, PrintedinGreat Britain
STABILITY OF A PIPE CONVEYING PULSATILE FLOW JERRY
A
H. GINSBERG
Purdue University, School of Mechanical Engineering, West Lafayette, Indiana 47907, U.S.A. West Lafayette, Indiana 47907, U.S.A. Abstract-This investigation concern the small displacements of a pipe conveying a pressurized flow whose velocity possesses a harmonic fluctuation about a mean value. A new derivation of the fluid forces is used to obtain the partial differential equation of motion. General equations applicable to any set of boundary conditions are specialized for the case of a simply supported pipe and the Galerkin method is utilized to find solutions. An analysis of the case of steady flow shows that the pipe exhibits the divergence type of instability which is predictabie by static structural theory. In the presence of pulsatileflow the pipe has regions of dynamic instability whose extent increases with increased amounts of fluctuation, The results have great similarity to those for beams carrying pulsating end forces.
INTRODUCTION THE WBRATION
and stability of pipes conveying flowing fluid fiows has been the subject of a long series of investigations, see [l-9] for example. These studies have considered the effects of fluid velocity, pressure, viscosity, and compressibility upon both infinitely long pipes and pipes with various types of boundary conditions. However, in each case the equations of motion were solved for the case of a fluid flowing at a constant speed. This study was motivated by the fact that it is not likely that a pump will yield a truly constant velocity. In fact for some pumps the velocity fluctuation is quite appreciable. The problem investigated herein concerns the parametric excitation of the transverse displacement of a pipe due to a pulsating flow. This results in the possib~ity of an oscillatory instability at mean flow velocities below the critical one for steady flow. In general, transverse motion of the pipe will affect the character of the flow, but it is assumed that this effect is negligible. The fluid is considered to be incompressible and inviscid with a uniform velocity profile. The pipe has a constant cross-section which is completely filled by the fluid. There is a pump at the end s = 0, and the end s = L is connected to a chamber which is maintained at a constant pressure pO. General equations of motion for small transverse displacements are derived for any set of end conditions, and solutions are obtained for the case of simple supports. Although the equation for the transverse force exerted by the flowing fluid upon the pipe has been derived elsewhere, a new derivation is presented here. It is believed that this derivation will prove to be extremely useful in an analysis of the effects of large transverse displacements upon the predicted regions of dynamic instability, EQUATIONS
OF MOTION
Let the pipe be in a deformed position. Consider a segment of fluid of length ds at a d$$tnn_ces from the pump. Let XYZ be a space fixed reference frame with unit vectors I, f, K and let xyz be a reference frame attached to the pipe with unit vectors zx 2, as shown in Fig. la. The acceleration of the fluid is given by 1013
IJEsVd
11 No9-F
1014
JERRY
H. GINSBERG
If the diameter of the pipe is sufficiently small compared to the length, then Go will be negligible. Thus, letting u and w be the Lagrangian components of the displacement of point 0, the remaining terms in equation (1) are
Fig. la. Coordinate systems and geometry for a deformed section of pipe and fluid.
Fig. 1b. Free body diagram of an elemental slug of flowing fluid.
The dynamic stability of a pipe conveying a pulsatile flow
When jw/L( Q 1, cos r#~= 1, sin 4 = 0, ?-
1015
i, c== f?, Z = 0, 4 = awl&, and l/R =
a%/as? Now consider
a free body diagram of the slug of fluid, Fig. lb. Equilibrium in the z direction requires that the transverse force per unit length _/be given by (3) where p is the density of the fluid and A is the internal cross-sectional area of the pipe. Substituting equations (1) and (2) and applying the conditions for small displacements then yields (4) The velocity and pressure of the fluid are related to the output of the pump by the laws of fluid dynamics. The pump produces a harmonic pulsation in the flow velocity, and the open end of the pipe is at a gauge pressure p. z+((s=O,r)
= vo+v,sinot (9
p(s=L,t)
=po.
The continuity condition for the incompressible
inviscid fluid requires that
Q(S, t) = ~(0, t) = vo+ u1sin ot
(6)
while equilibrium tangent to the axis of the pipe requires that
ap p!% -=as
at.
(7)
The solution of equation (7) which satisfies equations (5) and (6) is p(s. t) =po+pov,(L-.s)
cosot.
(8)
Substitution of equations (6) and (8) into equation (4) then yields an expression for the transverse force. It is assumed that the material of the pipe obeys a stress-strain relationship of the Kelvin type, i.e.,
In the case of small transverse displacement pipe may be obtained from [ 1 I]
the equation for dynamic equilibrium of the
(10) where I is the second moment of area and m is the mass per unit length of the pipe.
1016
JERRY
For convenience
H. GINSBERG
define the following non-dimensional
e=$
7 =
p=e,
e=d,
f&t,
parameters:
y=E.L, 1
(11)
+y,
a=PA
pA+m
where Oj is the jth natural frequency of the pipe when the fluid is at rest and unpressurized. In the case of a simply supported pipe
1* l/2
wj
=
j2f12
(pA?m)L4
After performing the necessary substitutions
‘+ (pA
(12)
the equation of motion becomes
+:)LaO;
(wi”+GwiL’)+[cQZ(1+2ysin&+y2sin2&) +~+o~~e(l-~)cos~~]w”+2o~(l+ysin8~)~’=0.
(13)
A dot and a prime indicate differentiation with respect to T and 5, respectively. To determine the response the displacement w is expanded in an infinite series of the normal modes of free vibration of the pipe when the fluid is at rest and unpressurized w =
2 ~j(()uj(T);
c#q- (pA+Ey4+#Jj = 0.
(14)
j=l
For the case of a simply supported pipe +j({) = sin j9r[.
(15)
Now substitute equation (14) into equation (13), multiply the result by c#Q(~)(i is any positive integer), integrate over 0 S 4 S 1, and use the orthogonality condition for the modes. The result is &+~(&&+a,)+20$(l+ysin&) 4
i
pijcij
j=l
+[a/3z(l+2ysin&+y2sin2&)+~l
2
qijaj
i=l
+
apye
cos
87 i j=l
where
r{jUj
= 0;
i= 1,2,.
. .
(16)
1017
The dynamic stability of a pipe conveying a pulsatile flow
(17)
It is obviously not possible to work with equation (16) unless i andjare truncated at some finite value, resulting in a solution by the Galerkin method. For the simply supported pipe, the Euler buckling modes are identical to the modes of free vibration. Thus, a two term truncation will result in stability criterion which are reasonably accurate for instabilities associated with the first mode, and less accurate for instabilities associated with the second mode [ 123.In this case equations (16) and (17) give ~,+6h,-~~p(l+~sin8~)d,+[~~-~201p2y(2sin6)7+ysin2~7) +27rz+
l)arprecos &-la, -?&pye(cos
&)a, = 0
The values a2, and a2, correspond to the first two natural frequencies of a pipe carrying a steady flow obtained from an analysis which neglects the effect of Coriolis acceleration.
sz,= [1 -+(ap”+&&)]“z (19) 112
a,=4
1-$ap’+&&)
.
I
I
It should be noted that for the simply supported pipe pfj = 0 if i = j, so that the effect of the Coriolis acceleration of the fluid would be unaccounted for if only one term in equation (14) was retained. STEADY
FLOW
When the velocity of the fluid with respect to the pipe is constant, y vanishes. The equations of motion (18) then reduce to a pair of coupled ordinary differential equations with constant coefficients. Their solution is q(7)
= i
aijeV;
.i=i
i=2,2
(201
where the values A,are the roots of the characteristic equation h4+ 17SA3+(n:+@-t-
16i32-t-~~2j32)h2+ (16~::+~,2)SA+Q~~n,2= 0.
(2 1)
1018
JERRY
H. GINSBERG
The solutions (20) are stable if Re (Xj) c 0 (j = 1 - 4). Because the coefficients of equation (21) are real, the roots always occur as two pairs of complex conjugates. First consider the situation when damping is absent, 6 = 0, in which case equation (21) is biquadratic. In order for the response to be stable it must be true that both values of A2be negative. This criterion is potentially satisfied in two regions. The first region corresponds to the case 0; > 0, which is satisfied if p < p* where p* =
[;($-1*)]1’2.
(22)
Any value of p exceeding that of equation (22) corresponds to the static instability of divergence, previously investigated in reference [6]. If p > l/,rr* the response is always statically unstable. It is also possible for A*to be negative for values of /3 slightly greater than that which gives fli = 0,
P>
[-$qp.
(23)
This situation occurs whenever the coefficient of A* in equation (21) is positive for fii = 0. Substitution of equation (19) reveals that this small region of dynamic stability is obtained whenever 27 k
4a
(24)
A typical case is illustrated in Fig. 2. The listed values of the parameters approximately represent a 15 ft, 6 in. diameter steel pipe with O-05 in. wall thickness when p. = 100 p.s.i. Divergence occurs at /3 = 0.338 and the small region of dynamic stability occurs for 0.703 6 p G 0.741. It should be noted that for two reasons this latter region is of academic interest only. First, as noted previously it is statically unstable, and second, it disappears when even a small amount of damping is present in the system. When 6 # 0 the two pairs of complex conjugate roots of equation (21) always have non-zero real parts, except when CIf = 0 (p = /?*> or fit = 0. Stability requires that each root satisfy Re (A) < 0, which is found only for p < p*. The topology of the graphs of lZm(A) ) vs /3 and Re (A) vs j3 is essentially the same as when 6 = 0, except for values of /3 greater than that which gives fi22= 0. In this region the value of Re (A) always bifurcates upward from the point Re(A) = 0, so that one branch of Re(A) always corresponds to an unstable solution. In Fig. 2 the values of (Im (A) ( and Re(A) are compared for the case 6 = 0 and for a typical small value 6 = O+IO5. PULSATING
FLOW
When the flow is unsteady (y # 0) general solutions of equation (18) are not available. However, here we are concerned only with the question of whether the undeformed position is stable in the sense of Liapunov, i.e. is ai bounded for all values of 7. Using Floquet’s theory [13, 141 it is easy to show that since the coefficients
The dynamic stability of a pipe conveying a pulsatile flow
1019
-0,
-0,
4-
-0
RdXkJ
3-
\ 1; e4 E L:
--(
\
--___-8-0 z-
-6=om5 --
\
S=O and 6=0.005
I
\
--/
roc “4
I\
Im(X)
\
\
PC non-dimensional
im(X)\j \ I \
fluid velocity)
Fig. 2. Roots of the frequency equation for steady flow when o! = 0.8 and p = 0.01.
of equation (18) are periodic in T = 2?~/0,transition from stability to instability will be marked by the existence of solutions which have a period of T or 2T. Thus, the values of the parameters which define the stability boundaries are obtained by seeking conditions under which such periodic solutions exist. Following the method described in [12] solutions whose period is 2T may be constructed from the Fourier series CZt(7)= i
j=l
[bij COST+
Cij sin’+] ;
i=
1,2
(2.5)
while solutions whose period is T may be constructed from the Fourier series a, (7) = 4dfo-I-2 [du cosj&+fij
j=1
sinj&J;
i= 1,2.
(26)
In the limit as p + 0 (slow fluid velocity) regions of instability are found to exist in the vicinityofo=2wIMando=2&j, (j--1,2,.. .). The regions corresponding toj = 1 are called the principal instability because they have the widest extent and also because they are least affected by damping. This study will be confined to investigating the p~ncipal region of instability and also the first region of secondary instability (.j = 2).
1020
JERRY
H. GINSBERG
If the value of p is sufficiently small compared to p* (i.e. if the steady component of fluid velocity is not too close to the divergence condition) reasonable accuracy in the determination of the principal region will be obtained by truncating equation (25) atj= 1 ~~(7) = 6,, cos$+c,,
sin:;
i= 1,2.
(27)
This expression is substituted into the equations of motion (18) and the method of harmonic balance is used to obtain simultaneous homogeneous equations for the coefficients bil and cil. For a nontrivial solution the determinant of their coefficients must vanish jkij[ = 0
(28)
where 62
1 1)&O--~p*@y*
1
k,, = fl,z-,$2.rr2+ k,, = tti0 -
rr*c@*y
k,, = -k,,
= -6aPy9
k,, = -k,,
= -k,,
= k,, = -$a@’
k,, = -883 - IT*c@“~
(29) k,, = -k,,
= -F&0
k,, = &&$r*+
l)a~yO-27r2a/32y2
k,,= 888 - 4r2a/3*y k43 = -860 - 41~*01~~y
k,, = a;-;+$(87r*+
Y +
1)a/3yO-27r2~~2y2.
For the case without damping, evaluation of equations (28) and (29) in the limit 0 yields (k,, k,, - k,“, ) (k,, k,, - kf4 ) = 0. Both factors give the same result.
(g4-(ng+n:+~a*~*)(~)2+n~~~ =0.
(30)
Comparing the above with equation (2 l), it is seen that (8/2) * = -A*. Therefore, twice the frequencies of free vibration for the steady flow case when /3 < /3* form the back-
The dynamic stability of a pipe conveying a pulsatiie
1021
flow
bone curves for the regions of the principal ins~b~ity. It will be seen that y controlis the width of these regions, The secondary regions of instability may be studied in a similar manner. Again for sufficiently small p, equation (26) may be truncated atj = I U,(7) = frdio+ d,, cos & +.f& sin 67.
(31)
The above is substituted into equations (18) and six homogeneous equations for the coefficients are obtained by the method of harmonic balance. The boundaries of the regions of secondary instability are then obtained by solving for the values of 8 which cause the dete~inant of the system of equations to vanish. It is found that when damping is not present the limit y + 0 yields e4-- (n~+n,z+yay32)e2+Qg2;
= 0
(32)
so that the backbone curves for the regions of secondary instability are the same as the frequencies of free vib~tion for steady Bow when j3 < ,@*.Extending this result it may be shown that 2 [1m(h)l/jform the backbone curves for thejth regions of instability. In general the principal regions of instability are of greatest concern, They have the greatest width and are thus most likely to occur, as is shown in Fig. 3. A study of the
______
s 86
850
--
2
23
a=002
6=Oond
6=0.02
6 2 5 s0 4 c 0
______
----^-Lc---~-
-__-<====----e_ - _ _ _ _ _==-_-_-=<
.-s F
E
0
.&2 t? 8 ___c_L_--_____________
-~ 0
1
-vm /
005 p
---==-==X~ I O-15
1
51
(non-dimensional
fluid
/ 0.2
J 0.25
velocity)
Fig. 3. The effect of damping on the stability boundaries for pulsatile flow when (Y= 0.8, /.L= 0.01 and y = 0.4.
1022
JERRY
H. GINSBERG
effect of damping shows that when 6 = 0 there are values of 0 representing unstable responses for every value of p, whereas when damping is present p must attain a critical value in order for any instability to occur. This critical value increases as 6 increases. Furthermore, damping in general has greater effect upon regions of instability associated with the higher natural frequencies and also upon higher order regions of instability. In fact, for the example of Fig. 3, the secondary region associated with the second natural frequency has disappeared. The effect of different amounts of pulsation is illustrated in Fig. 4, wherein it is seen that a 100 per cent fluctuation (y = 1) results in relatively wide regions of instability and a 10 per cent fluctuation results in narrow regions. For brevity diagrams illustrating the effects of each parameter are not presented. Noting that the backbone curves for each region are described by equations (30) and (32), it was found that the main effect of increasing the pressure (increasing CL)was to decrease the scale of the ordinate 0 and absisca /3; lowering the density of the fluid (decreasing (u) increased the scale of the absisca /3 only. DISCUSSION
The investigation of a pipe conveying a pulsating fluid flow has shown that for values of mean flow velocity that are small compared to the one causing static instability
-L 025 p (non-dimensional
fluid
velocityl
Fig. 4. The effect of the amount of pulsation in the flow velocity on the stability boundaries when a = 0.8, w = 0.01, and 6= 0.02.
The dynamic stability of a pipe conveying a pulsatile flow
1023
(p < p*), there is little chance of an ins~bility occurring for two reasons. First, because all systems have some damping, regions of instability exist only if j3 exceeds a certain critical value. Second, even if /3 exceeds this value, the range of frequencies which correspond to unstable situations is extremely limited for the usual amounts of velocity fluctuation. As the value of fl is increased the range of unstable frequencies widens, so that dynamic instability becomes more possible. However, it should be remembered that for higher values of j3 additional terms in the time series for a, (7) and a2 (T), equations (25) and (26), must be retained. Thus, in this case Figs. 3 and 4 must be used with care. Also, the regions of instability associated with the second natural frequency are less accurate than those associated with the fundamental frequency, because the higher modes were not retained in the analysis. A comparison of the results for the pipe to those for the problem of a beam carrying a pulsating follower force1121 deserves consideration. The forces exerted upon both structures are nonconservative. In the absence of fluctuation in the value of the end force or flow velocity, instability for structures that are simply supported may be predicted by a static analysis, whereas for other types of supports instability can be correctly predicted only by a dynamic analysis. When pulsations are present, the nonconservative aspects of the forces merely modify the extent of the regions of instability and do not cause any new phenomena. Furthermore, the diagrams of the regions of instability for the two problems have very similar patterns. Interesting topics for continuing study are the effects of fluid viscosity and turbulence. Viscous effects were shown to be unimportant for steady flow [4,7], but it is not clear that this will be true for pulsatile flow. This study can be considered to describe the qualitative effects of turbulence, but a more accurate study of this matter would require solution of the equations of motion when the time dependent coefficients are stochastic, rather than harmonic, functions. REFERENCES 111 H. ASHLEY and G. HAVILAND, 1. uppl. Mech. 17,229 (1950). 123 G. W. HOUSNER, J, uppI. Me&. 19,205 (1952). [3] F. I. N. NIORDSON, Trans. R. Inst. Tech. 73(1953). [4] T. B. BENJAMIN, Proc. R. Sot. Ser. A, 261,457 and 487 (1961). 151 V. W. ROTH, Ing.-Archiu33,236 (1964). f61 H. L. DODDS, Jr. and H. L. RUNYAN, Technical Note D-2870, National Aeronautics and Administration (1965). [7] R. W. GREGORY and M. P. PAIDOUSSIS, Proc. R. Sot. Ser. A, 293,512 and 528 (1966). [SJ S. NEMAT-NASSER, S. N. PRASAD, and G. HERRMANN, AIAAJ. 4,1276 (1966). [9] S. NAGULESWA~N and C. J. H. WILLIAMS, J. me&. Engng Sci. 10,228 (1968). 1101 R. A. STEIN and M. W. TOBRINER, 1. uppE. Mech. 37,906 (1970). [l I] W. NOWACKI, Dynamics ofElastic Systems. Chapman & Hall (1963). [12] V. V. BOLOTIN, Dynamic Stability of Elastic Systems. Holden-Day (1964). ]13] R. A. STRUBLE, ~oali~eur Partial Dz~ere~r~afEquations. McGraw-Hill (1962). ii41 L. MEIROVTICH, Methods ofAnalytical Dynamics. McGraw-Hill (1970).
Space
(Received 24 October 1972) R&unt!-Cette recherche conceme les petits deplacements d’un tube conduisant un ecoulement sous pression dont la vitesse presente une fluctuation harmonique autour d’une valeur moyenne. Une nouvelle derivation des forces du liquide est utilisee pour obtenir l’iquation aux dilferentielles partielles du mouvement. Des equations g&&ales applicables a tout ensemble de conditions aux limites sont adapt&es au cas d’un tube simplement support& et h methode de Galerkin est utilisee pour trouver des solutions. Une
1024
JERRY
H. GINSBERG
anaiyse du cas de I’icoulement stable montre que le tube pr&ente le type d’instabilite de la divergence, que l’on peut prevoir par une theorie structurelle claasique. En presence d’un ecoulement pulsatoire, le tube a des regions d’instabilite dynamique dont l’itendue s’accroit avec une augmentation de la quantite des fluctuations. Les resultats ont une grande similitude avec ceux des poutres supportant en leur extritmite des forces pulsatoirea. Zusammenfassung- Diese Untersuchung betrifft die kleinen Verdr%gungen eines Rohres, das einen unter Druck stehenden Strom fdrdert, dessen Geschwindigkeit eine harmonische Schwankung urn einen Mittelwert aufweist. Es wird eine neue Ableitung der Fl~ssigkeitskr~te verwendet, urn die partielle Differentialgleichung der Bewegung zu erhalten. Allgemeine, auf jeden Satz von Grenzbedingungen anwendbare Gleichungen werden auf den Fall eines einfachen unterstiitzten Rohres spezialisiert und die Galerkin’sche Methode wird zum Auffinden von Lijsungen verwendet. Eine Analyse des Falles stetigen Stromes zeigt. dass das Rohr den Divergenztyp von ~nstabil~t~t aufweist, der durch statische St~ktu~heo~e vorhersagbar ist. in Gegenwart von Pulsierstrom hat das Rohr zonen dynamischer Instabilitat, deren Ausdehnung sich mit vergriisserten Schwankungsmengen vergrossert. Die Resultate sind den Resultaten fiir pulsierende Endkr%fte tragende Balken sehr Ihnlich. Sonmario - Quest0 studio verte sui piccofi spostamenti di un t&o the trasporta un flusso pressurizzato la cui velocita ha una fluttauzione armonica intomo a un valore medio. Una nuova derivazione delle forze fluide viene impiegata per ottenere I’equazione differenziale parziale del moto. Si specializzano equazioni generali applicabili a qualsiasi gruppo di condizioni limite per il case di un tubo a semplice supporto e si usa il metodo di Galerkin per trovare le soluzioni. Un’analisi de1 case di flusso unifo~e dimostra the il tubo mostra il tipo di divergenza d’instabilita the & prevedibile mediante la teoria strutturale statica. In presenza del flusso pulsatile il tubo ha regioni d’instabilit8 dinamica il cui valore aumenta con I’aumento nella fluttuazione. I risultati si avvicinano molto a quelli delle travi the portano forze assiali pulsante. t%CTpWTGoo6maer 0 He6OnbUILiX CMeu&HKRX Tpy6bI B CnyYW, KOrAa CKOpOCTb lIOTOKa.npH ROBbIIlIeHHOM AitBJleHBH BMeeT rapMOHHYeCKOe Kone6ame OTHOCHTeJIbNO CpeAHerO 3HaYeHWI. Ha OCHOBe HOBOPO BbIBOAa H(HAKOCTHblX CUII llOJlyYaeTC8lAH~&WfWiaJIbH~ ypaBHeHKe ABEKeHMJl B YaCTHblX npOH3BOAHbIX. &PI uOMep~U~HH0~ Tpy6bI OCO6eHHO EOIIyYZUOTCSlo6mne ypaBHeH&iR lIpUMeH%iMb!e K npO~3BOAbHO~ cucfetde rpaH~YHb~X ycnomi%, peruembie npE rioMouw MeTaDa RmepKmia. @is aHami3a cny%w CTauwoHapHoro I'lOTOKarpy6a npOflBlI~eT yCTOi%lBOCT% JWiBepreHTHOrO Tllna, YTO B03MO?KHO noToKa npe.IWKa3aTb, liCIIOJIb3ySI CTaTHYeCKyH, CTpyKTypHyHJ TeOpUIO. IIplr HamiYmi nynbcEcpymuer0 Tpy6a &iMeeT o6nacTu AEiHaMllYeCKOi%)'CTOk'ftiBOCTIi, IIpOTSDKeHEieKOTOpbIX yBeJ&iYaeTCffC yBeJEiYeHHeM Kosre6~uii.rrpu3TOMahAeeTCA3HaMeSIITenbHOe nono6we Me~Ay3T~M~~3YAbTaTaM~~TeM~ AJIRlQ”F3G HeCy3IJEiX IIyJIbCSipyIoWue KOHeYHbIeCIIJIbI.