Download Correction for housner`s equation of bending vibration of a pipe line

Applied Mathematics and Mechanics
(English Edition, Vol. 14,No.2,Feb. 1993)
Published by SUT,
Shanghai, China
C O R R E C T I O N F O R H O U S N E R ' S E Q U A T I O N OF B E N D I N G V I B R A T I O N
OF A PIPE LINE CONTAINING FLOWING FLUID
Zhang Xi-de (~,~,~,,)
Du Tao ( ~
~)
Zhang Wen ( ~
3~)
(Qingdao Institute of Chemical Technology, Qingdao)
Shen Wen-jun(~3~ ~ )
(Xi'an Jiaotong University. Xi'an)
(Received March 23, 1990; Communicated by Chien Wei-zang)
Abstract
Abstract
This paper points
out that
Housner's
bending
vibration
a pipe line
The one-dimensional
problem
of the
motion equation
of a rigidofflying
plate
under of
explosive
attack has
an analyticcontaining
solution./7owing
only when
the
polytropic
index
of
detonation
products
equals
to
three. In
fluid is approximate and makes correction to it. An exact form of
general, a the
numerical
analysis
is
required.
In
this
paper,
however,
by
utilizing
the
"weak"
shock
vibration equation is given.
behavior of the reflection shock in the explosive products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying
words
induced vibration,
feedback,
velocity
plate driven byKey
various
high explosives
with polytropic
indices
otherfield,
than potential"
but nearlyfunction,
equal to three.
principle,
couple
vibration
Final velocities of flying plateHamilton's
obtained variance
agree very
well with
numerical
results by computers. Thus
an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic
I.
I n for
t r o estimation
d u c t i o n of the velocity of flying plate is established.
index)
The analysis of dynamics and stability of a pipe line containing flowing fluid is very
important both in theory and in engineering
practice. From the beginning of the 50's, a lot of
1. Introduction
studies have been done on it. The induced-vibration differential equation of a pipe line, which
Explosiveextensively,
driven flying-plate
technique
ffmds its
important
use in the study of behavior of
was accepted
was given
in Housner's
paper
tt~ as follows
materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
04y . of estimation
8Zy
cladding of metals. The method
of flyor8Zy
velocity and the waydZy
of raising it are questions
EI-~'+'2pAv'8--'~-]-pMv z ~
-I-(mp-l-pA) --~T- = 0
('1 . I)
of common interest.
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
E: modulus
of elasticity
of pipe
approach
of solving
the problem
of motion of flyor is to solve the following system of equations
I: moment
offield
inertia
of pipe products behind the flyor (Fig. I):
governing
the flow
of detonation
A: internal cross-section area of pipe
rap: pipe mass of per unit length
--ff
=o,
ap +u_~_xp+ au
p: tiuid mass of per unit length
au
au
v: fluid velocity
y1
=0,
Equation (1.1) is usually called Housner's equation. Many analyses have been made(i.0
based
aS bya sthe velocity of the flowing fluid in a tube was
on it. But only the vibration induced
a--T
=o,
considered, the reaction of the tube to the fluid was neglected. Therefore, Housner's equation
p =p(p,
which did not consider all the aspects
of thes),
vibration is an approximate one.
In this paper, an exact equation of bending vibration of a pipe line containing flowing
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
fluid
is obtained
studyingRtheofreaction
the pipe
on the fluid.
respectively,
with after
the trajectory
reflectedofshock
of detonation
wave D as a boundary and the
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it are governed by the flow field I of central
159 rarefaction wave behind the detonation wave
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
293
160
II.
Zhang Xi-de, Du Tao, Zhang Wen and Shen Ken-jun
D e r i v a t i o n o f E x a c t V i b r a t i o n D i f f e r e n t i a l E q u a t i o n o f a Pipe Line
The system under consideration consists of a uniform tubular beam of length l, internal
radius a, conveying uncompressible fluid of density p, flowing axially with a constant velocity
o 9 It is assumed that the bending vibration of the pipe is one of a beam, and the disturbed flow
of the fluid in the pipe, arising from the vibration of the pipe, is potential. According to
irrotational motion theory of ideal fluid, a potential function exists in the velocity field of the
disturbed flow. Letting it be ~ ( r , 0, x, / ) , it should satisfy Laplace equation.
Corresponding to a pipe line having determined boundary conditions, the concrete expression
of the potential function ~ can be derived.
Assume that ~ ( r , O , x , t )
is given the add fluid-dynamic-pressure at any point on the
inner wall o f the pipe, by use of Cauchy-Lagrange integral in ideal fluid dynamics, is
calculated by
Abstract
O~
P-~ - P Ot
The one-dimensional problem of the motion of a rigid flying plate under explosive attack has
an analytic
solutionononly
the polytropic
index of force
detonation
equals to three. In
So,
the projection
the when
v-direction
of the resultant
of theproducts
add fluid-dynamic-pressure
general,
a
numerical
analysis
is
required.
In
this
paper,
however,
by
utilizing
the "weak" shock
per unit axial length is
behavior of the reflection shock in the explosive products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying
1)] ~=,=with
- P polytropic indices
~cosOdO
(2.1)
plate driven by various high explosives
other than but nearly equal to three.
r=a
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
Evidently,
reaction
of the
vibration
of explosive
the pipe (i.e.
walldetonation
is equivalent
add axial
an analytic the
formula
with two
parameters
of high
velocitytoandanpolytropic
distributed
pressure of
acting
on the of
inner
wall
of is
theestablished.
pipe.
index) for estimation
the velocity
flying
plate
Under small amplitude vibration condition, there is relation d v / d x < ( 1 , where V
represents cross displacement of the 1.pipe;Introduction
x-represents axail coordinate. So, the horizontal
component and the vertical component of the fluid velocity in the pipe are
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
Oy. vof Oy
materials under intense impulsive loading, shock synthesis
diamonds, and explosive welding and
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
of common
The
kinetic interest.
energy and the potential energy of the pipe system are expressed by
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
approach of solvingTthe
problem
motion of flyor is to solve the ofollowing system of equations
=_~_It0
{ of[dY\z
governing the flow field of detonation products behind the flyor (Fig. I):
v=
o
--ff
=o,
- - P I , += a ' Vau
ap +u_~_xp
au
au
aS
as
1
y
=0,
respectively. According to Hamilton variational principles
Ill a--T
yields:
p
(i.0
=o,
=p(p, s),
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
f,," with the trajectory R of reflected shock of
- detonation wave
+PlD,~a"
} d x d t = Oand the
respectively,
as avboundary
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters
on integral,
it are governed
by the boundary
flow field conditions,
I of central rarefaction
By
partial
considering
there is wave behind the detonation wave
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
293
Correction for Housner's Equation
161
~ii Jo(
f'~EIO~:Y
aZY pay zO~Y
O2y
ox w 2pAvo--~+
-~-~-+(mr+pA)-~i~+pl,.o)dydxdt=O
Since 3y is arbitrary, it is obtained
E- O*y .
OZy
0z
.. + (
(2.2)
,+pA)-Tir-+p],..=
Substituting (2.1) into (2.2) yields
o'y
or I
E I-O--~x~
+ 2pAv-ff-x~+ PAu2 --~-f'xz+ (m' + p A ) 7o:y - P 3o --~-I
,.." acosOdO~ 0
(2.3)
Eq. (2.2) or Eq. (2.3) is the exact vibration differential equation of a pipe line conveying fluid.
Compared with Housner's equation, it adds the fifth term on the left, that is, the correction
Abstract
term for Housner's equation.
Usually,
the potentialproblem
function
is one
of the
displacement
of the
The one-dimensional
of the~(r,O,x,t)
motion of a rigid
flying
platevibration
under explosive
attack has
pipe,
and expressed
in integral
form,
Therefore,
Eq.of(2.2)
or Eq. products
(2.3) is aequals
differential
in'tegral
an
analytic
solution only
when the
polytropic
index
detonation
to three.
In
equation.
general,
a numerical analysis is required. In this paper, however, by utilizing the "weak" shock
behavior of the reflection shock in the explosive products, and applying the small parameter purReference
terbation
method, an analytic, first-order approximate solution is obtained for the problem of flying
plate driven by various high explosives with polytropic indices other than but nearly equal to three.
[1] Housner, G.W., Bending vibration of a pipe line containing flowing fluid, J. Appl.
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
Mech.,
June (1952),
an analytic
formula
with two205.
parameters of high explosive (i.e. detonation velocity and polytropic
index) for estimation of the velocity of flying plate is established.
1.
Introduction
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
of common interest.
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
approach of solving the problem of motion of flyor is to solve the following system of equations
governing the flow field of detonation products behind the flyor (Fig. I):
--ff
ap +u_~_xp+
au
au
aS
as
au
y1
=o,
=0,
(i.0
a--T
=o,
p =p(p, s),
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
293