Dynamic stability of a liquid carrying pipe

MECHANICS RESEARCH COMMUNICATIONS 0093-6413/79/030141-06502.00/0

Vol. 6(3),141-146, 1979. Printed in the USA. Copyright (c) 1979 Pergamon Press Ltd

DYNAMIC STABILITY OF A LIQUID CARRYING PIPE A. Tylikowski Institute of Machine Design Fundamentals Warsaw Technical University, Poland (Received 10 January 1979; accepted for print 16 April 1979)

Introduction The stability of pipes conveying incompressible inviscid fluid has been considered in literature for the last twenty five years (e.g. ~,2,3]). Early works concerned the effect of steady flow and internal pressure on natural frequencies of pipe as well as the critical flow velocit~ at which the pipe lost stability. These papers have considered the beamlike behavioz of the straight pipe. More recently interest has developed in stability of pipe considered as cylindrical shell [4,5~and uniformly curved tubes [2 • All of these papers have applied a finite-dimensional approximation in stability analysis. Using the L i a p u n o v - M o v c h a n method the stability of a straight hose laying on elastic foundation conveying fluid was recently investigated 7]. Most recently Holmes treated the stability of all equilirium positions of pipe conveying fluid using the same method and taking into consideration a structural nonlinearity [8], The present paper reports a study of stability of the fluid carrying straight pipe laying on elastic foundation. Destabilizing effects of liquid motion as well as stochastic component of an axial force acting on the pipe are taken into consideration. The direct Liapunov-Movchan method is used to derive the sufficient stochastic stability conditions. Therefore the flexible hose with constant axial force and the liquid conveying pipe without elastic foundation are analysed as particular cases

~

Basic Equations And Defin%'tions We consider a straight uniform pipe of constant both cross-sectional area and area moment of inertia and laying on elastic foundation. The pipe is subjected to an axial force as well as transversal hydrodynamic force caused by liquid motion with constant velocity inside the pipe.Assuming plane displacement state 141

142

A. T Y L I K O W S K I

of the ces by motion may be

pipe axis and determining the inviscid hydrodynamic forslender body theory the linear equation of transversal of the pipe conveying the inviscid incompressible liquid written as follows B~u ~v + ~ v + ru + EpJ ~ _ (~tl + m2) ~ v + 2zm2 a~ :IT O + T(t) - pA - m2 z2] ~ ,

~g

(0,i),

CI)

- 5

where mfl = mass of pipe per unit lenth z = flow velocity A = cross-sectional m 2 = mass of liquid inside the pipe area per unit length r = elastic constant of foundation Ep= You~'s modulus J = area moment of inertia = damping factor p = pressure inside the pipe t = time 1 = length of pipe T O = constant component of axial force = axial pipe T(t)= stochastic component coordinate of axial force ~u v = ~ - = velocity u = transversal displacement measured from straight state of pipe Assuming simply supported edges we have the following boundary conditions u(O) = u(1) = 0

'

~(0) a§2

'

-

~2u(1) : 0

~)~2

Introducing dimensionless variable x =~/I the equation of motion (I) in the form V + 2 w '~t

V t

+ 2 b v + c u + e u""-[fo

"

(2)

we can rewrite

+ f(t)] u" = O, x6(0,I)(3)

where the prime indicates partial derivative with respect to dimensionless variable x and b = ~ / 2 ( m I + m2)

w = z m2/ l(m I + m2)

fo=(To - pA - m2z2)/ 12(ml + m 2)

e = EpJ/ l~(ml + m 2)

f(t~= T(t)/ 12(m I + m2)

c = r/(mfl + m2)

The equation (3) with zero initial conditions u(x,O)= O, v(x,O)= 0 possesses a trivial solution u(x,t)= O, v(x,t)= O. We shall investigate two concepts of the dynamic stabilit~ of

DYNAMIC

STABILITY

OF L I Q U I D C A R R Y I N G

PIPE

143

undeflected pipe axis, i.e. , trivial solution of equation (~ . First, under the assumption that the force f(t) acting in pipe axis is a stochastic stationary ergodic process we shall deal with almost sure asymptotic stability

In the second case, if the force is a Gaussian wide-band process (white noise) we shall analyse the uniform stochastic stability, i.e., we shall formulate conditions implying the logic sentence ~> 0

0

r>O

t> 0

where ~'II is a measure of the distance of solution of equation (3 from the trivial solution. Asymptotic Stabilit 2 Using ,,the best" functional constructed by the method of [9] we have

Kozin

v(O-(2b 2 + o)lu12+ %Iu'12+ elm"l% ~b(~,v)+lfl 2, where

I.I

and ( . , . )

denote the

~2

no=al

The functional (6) is positive-definite if condition)

and i ~ e r

(~)

product.

fo/e > _ ~ 2 (Euler's

v(0>~(b 2 + c)lu12+l~ + b~12+ (e~ 2 ÷ ~o) I~'I ~ ~o. Upon differentiating (6) we obtain dV = 2 [(2b 2 + c) (u,v) + fo(U',V')+ e(u",v")+ dt

~I~12+(~+ b~,~]

Substituting (3) integrating by parts and using the boundary conditions (2) the time-derivative becomes dV dt = - 2bV + 2U, where

U

is the functional

(7)

144

Ao T Y L I K O W S K I

We apply the variational calculus to obtain a function k satisfying inequality as follows U.< kV. (9) The associated Euler's equations of the problem are given by

b2u + f ( t ) / 2

u" + bwu' - A ~ b u + v) = O,

2bSu + b2v + f ( t ) / 2 2 +

(u - A v)= o

v" + f ( t ) b u " +

+

]

0o)

2bwv' : o.

Solving the equations 00~ with respect to ruction u and using the botuldar2 conditions (2) we find the appropriate function A as follows I/2 [ ~(t) = max An(t)= max,~ I[262 - f(t)(n~)2] 2 + %b2w 2 (n~D 2 .

Therefore, s u b s t i t u t i n g c o n d i t i o n (9) i n t o equation (7), r e s o l ving obtained differential inequality and assuming that stochastic process f(t) satisfies an ergodicity h~pothesis, which ensures the almost sure equality of time average and ensamble average, we derive the sufficient condition for almost sure asymptotic stability with respect to measure llull = V I/2, as E ~ < b, where E denotes an average operator. Hose With Zero Bendin~ Stiffness And Constant Axial Load Taking account of the foregoing assumption,we set e = 0 and f(t) = 0 in equation (3), functionals (6) and (8). Vanishing of the bending stiffness also causes a necessity of the assumption fo~O, implying the positive-definiteness of the functional fore ~ evaluated from (11) is constant function

(12) (6). There-

I/2 n=1,2, . .

DYNAMIC

STABILITY

OF L I Q U I D C A R R Y I N G

PIPE

145

Denoting D = b2(w 2 - fo) + w2c, we obtain an explicit form of constant A. If D ~ O , ~ is equal to b a n d if D < O , ~ is equal to A 1 . Thus we have the following stability conditions w2/fo
and

w2/fo>l

- 1/(I + b2/c),

and

w2/fo
- 1/(1 + b2/c).

(1@

or

w2/fo
+ C~fo~2)

Joining inequalities in the form

(14) and (15) we obtain stability criterion w2/f o < I.

(16)

But, in contrast to the general case, the differential inequality has the constant coefficient and we obtain a simple exponential estimation of functional V, the exponent of which depends on sign of D

V(t~V(O)[exp -2(b -X)t ].

We should notice that selecting the functional V in the form(6) allows us to make the stability criterion weaker than that obtained by Gutowskl in [7], who introduces additionally inequality as follows b 2 < c. Pipe Without Elastic Foundation In this case, c

is taken as zero in equation

(11)

X~)= max 7[.~b2- f(t)(n~2)2+ @b2~(n~J2]/[b2+fo(n~)2+ e('i~J[~]}1/2. ~1121

The above expression has the same form as that in the solution of stabilit-3 of panels in supersonic flow subjected to random end load [9], the Mach number being replaced by the quantit~ 2bw and unit bending stiffness. Un%"form Stabilit~ We rewrite the dynamic equation (3) in Ito differential form du = v dt, dv

=

(-2wv

(17) -

2by - cu + fo u'' - eu"") dt + ~ f

if the Gaussian wide-band force

f

U"

d~

can be approximated by time-

146

A.

TYLIKOWSKI

derivative of the Wiener process q with intensity Wf (white noise). Selecting the functional in the same form as in (6) apply Ito calculus to obtain its differential dV

we

dV : 2[(2b2+c)(u,du)+ fo(U', du~ + e(u",du")+ b(v,du)

+(v + b u , a v ) ] + ~ 2 {n'J 2 dr. Substituting the equations of motion (17), integrating by parts and using the boundary conditions (2) the differential dV can be written as follows

~ = -~b[ivl~÷o lull÷ ~o1~,1 ~ - ( ~

-e~l~" I ~]~t ÷~ ~(v÷~,u") ~

On integrating with respect to t from s to T;(t), where ~(t) = min(~,t), ~¢= inf{t: llull= ~ 0 } , averaging and taking into consideration that EO = 0 it follows that

[H ÷oI I ÷

e)I I

(18)

If the integrand in the equation (18) is non-negatlve we get inequalit-3 of the form EV(W~(t))~V(s), i.e. the functional is supermartingale. Neglecting term Ivl 2 and using Chebyshev's inequality to estimate an exceedence probability it follows that the sufficient conditions of uniform stochastic stability of undeflected pipe with respect to measure V 1/2 are fo/e ~ -K 2 ,

~2 ~2b

[e +(c + ~ 2

fo)/~$].

References 1. 2. 3%. 5. 6. 7.

Feodos'yev,V.P. ,Inzhenerayi Sbornik,10,169, (1951) • Benjamin,T .B. ,Proc .Roy .Soc., 261 ,%87, (1961) . Gregory, R .W., and Paidoussis, M.P., Proc .Roy. Soc., 293,528, (1966) Weaver,D.S..and Unny ,T .E. ,J.Appl.Mech. ,%0,%8, (1973) • Paidoussis,~.P. ,J.Mech.Eng.Sci. ,17,19, (J-~75) • Shoei-Sheno Chen,J.Acoust.Soc.Am.,51,223, (1972) • Gutowski,R.,Proc.XI Conf. Dynamics of Machines,Prague-Liblic e, p. 125, Academia, (1977) • 8. Holmes,P.J. ,J.Appl .Mech. ,~5,619, (1978) • 9. Kozin,F.,Lect.Notes in Math.,2~+,p.186,Springer, (1972). •