Idea Transcript
4
PARAMETRIC INSTABILITIES OF TUBES CONVEYING FLUID
by
NICOLAS
T.
ISSID
J
Under the supervision of Prof. M.P. païdoussis
A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the ~
requirements for
~e
degree of
Kaster of Engineering
1
Department of Me'chanical EngineetJ,nq
..
~
McGill University ,
"1.
oP,
., ,
Montreal, Canada
,.
(f-
.r .~
"~\
' - .. 't,
July, 1973
---
"
,
®
l
~~.~ ~ ~~
~
- ---------Rioolu T., INid
' ... Ji
1974
1
o ••
4
- i
•
-
ABSTRACI This thesis is concerned with the parametric response of tubes conveying pulsating fluid whose flow velocity has the form Uo(l +
~
cos wt), where
~
is small.
Boundary conditions
Il
corresponding to pinned-pinned, clamped-clamped and clampedfree constraints are considered. The equation of motion of a pipe conveying fluid is formulated to take into account unsteady flow, gravit y forces, and internaI and external dissipative forces. appr~ximate
Galerkin's
method is then ernployed which yields a set of
coupled Mathieu-Hill-type equations with rnultiharmonic coefficients.
The regions of instability are th en obtained by
following Bolotin's method to solve this set of equations. It is shown that the lowest order of approximation leads to results of fairly good accuracy for the cases of pinned-pinned and clamped-clamped tubes, whereas for a clampedfree one higher-order approximations are necessary (in~olving 20 x 20 characteristic determinant).
Also it i9
~h9wn
that
for a clamped-free tube the onset of instability is only possible ~
if the flow velocity and the excitation minimum values, even when
dissipa~ive
param~ter
forc~s
exceed certain
are absent.
The
effect of damping and coupling terms on instability regions
)
associated with different modes is discussed~.
~
f1F~es
resulting
The complete effect of the inertia
from the fluid axial acceleration on the behaviour of the tube ls
al~o onal;~ed (~was .
partially neglected in the past), and
"
a measure of the error involved by its neqlect i8 obtained.
- ii
~
INSTABILITÉS PARAMETRIQUES DES TUBES TRANSPORTANT DES FLUIDES
SOMMAIRE Cette thèse traite de l'excitation parametrique d'un tube par les pulsations du fluide qu'il conduit pour des vitesses d'écouleme~ de la forme Uo (1 + ~ cos wt), où ~ reste faible.
On considère successivement le tube comme étant à
support simple à chaque extremité, encastré à chaque extrémité et encastré d'un bout et libre de l'autre. La formulation de l'équation du mouvement du tube
.
transportant le fluide tient compte de
~'aspect
non-permanent
de l'écoulement, des forces de gravité et de forces dissipatrices
.
~
internes et externes.
~n
applique la méthode approximative de
Galerkin pour obtenir une série d'équations couplées, du genre de celles de Mathieu-Hill, contenant des coefficients multiharrnoniques.
La méthode de Bolotin, utilisée ensuite
pour résoudre cette série d'équation~, permet de déterminer les régions d'instabilité. On montre que le moindre degré d'approximation permet d'obtenir des résultats d'une précision passable dans le cas où le tube est à supports simples ou encastré aux deux
extr~t's~
tandis que dans les conditions encastré-libre, il faut avoir recours à des approximations d'un plus haut" degr& (entrainant . ,.~
un déterminant caractéristique de 20 par 20).
.
également que dans ce dernier cas
On' constate
l'instabilit~
prOduire que si la vitesse d'écoulement et le
ne peut se
param~tre
~
- iii -
d'excitation dépassent un certain seuil, mème en l'absence de forces dissipatrices.
L'ef'et qu'exerce l'amortissement d'instab~lité
ou le couplage de'termes sur les régions aux
diff~rents
modes de vibration
~st
réliées
aussi discuté.
Enfin on analyse au complet l'influence qu'exercent -
les forces d'accélération axiales du fluide sur le comportement du tube (analyse negligée en partie dans les travaux antérieurs)
-
et l'on évalue l'erreur introduite dans les calculs en négligeant
.
ces forces d'inertie.
, 1
/
\
-
"
f'
iv t'
(;1
ACKNOWLEDGMENIS !
The author wishes to express his gratitude to his supervisor Prof. /M.P. Paidoussis for his guidance and encouragement and the valuable suggestions.and criticisms which he has offered during the course of this work. the large amount/of
tim~
Also,
spent in reviewing the manuscript
is appreciated. The ,author is grateful to., the National Research Council of C~nada for their financial assistance (Grant
.
• A4366) and also for awarding him a Postgraduate Scholarship .
Sincere thanks are due t6 Mr. C. Guernier for his useful ideas during the editing of the thesis, and also to Miss'C. Leone for an excellent typing job. 1
1
1
, '\
- v -
TABLE OF CONTENTS 'ABSTRACT SOMMAIRE ACKNOWLEDGMENTS TABLE OF CONTENTS NOMENCLATURE CHAPTER 1
:"
i
i1
1v
v ix
INTRODUCTION
1
CHAPTER II DVNAMICS OF TUBES CONTAINING STEADV FLOW
5
2.1
Equation of Motion
6
2.2
Review of Steady-FZow Dynamios
8 o
2.2.1
CZamped-Free Tubes
2.2.2 •
CZamped-CZamped Tubes
10
DVNAMICS OF TUBES CONTAINING UNSTEADV FLOW
12
3.1
NathematicaZ ModeZ
12
3~2
Equation of Motion
14
8
CHAPTER III
-'l ...... ')~~
3.3
...... ,
PhysicaZ SimiZarity
19
CHAPTER IV
•
PARAMETRIC INSTABILITIES 4.1
4.2
Review of MathematicaZ Nethods
21 21
~SiS
25
4.2.1) primary Regions of Instabitity
25
4~,.2
29
Secondary Regions of InstabiZity
o
4.3
Distribut~on
of InstabiZity Regions
32
- vi -
e. CHAPTER V 35
THEORETICAL RESULTS 5.1
Region8 of In8tabiZity for a Pinned-Pinned • 35
Tube ~
5.2
f~r
Region8 of In8tabitity
a Clamped-Clamped
Tube
5.3
38
5.2.1 . Seleation of the Order of Approximation
38
5.2.2
Effeat of Flow Velocity
40
5.2.3"
Effeat of Corioli8 Foraee
42
5.2.4
Effeat of Di88ipative Forcee
..
43
Region8 of Inetability for a Clamped-Free 45
Tube
5.4
5.3.1
Se leation of the Order of Approximatitm
4'5
~. 3.2
Effeat of Flow Velocity
46
5.3.3
Effeat of
D~seipative
Foraes
49
Phyaiaal Interpretation of the Theo~etical Reaults
CHAPTER VI
.
/ , 50
, 53
REFINED THEORY
53
6.1
Introduction
6.2
Refined Equation of Motion
53
Theoret~cal ~ .
57
-6.3.1
Results t. Clâ-mped-Clamped Tube
6.3.2
CZamped-Free Tube
58
6.4
Conclusion
)
57
58
·S
- vii -
CH~PTER VII• 60,
CONCLUSION
"
REFERENCES
6!)
FI GURES 1 - 23
69
APPENDIX ~
A
..
,
THE CHARACTERISTIC DETERMINANT OF THE PRIMARY
A.I
REGIONS APPENDIX
B
,
THE CHARACTERISTIC DETERMINANT OF THE SECONDARY B.l
ReGIONS
.-
APPENDIX C THE CHARACTERISTIC DETERMINANT OF THE PRIMARY R~GIONS (REFINED THEORY)
C.I
APPENDIX D
\ "
THE CHARACTERISTIC DETERMINANT OF THE SECONDARY REGIONS (REFINED THEORY)
0.1
, APPENDIX4 E -', '
ANALYSIS FOR A DAMPED TUBE CONVEYING FLUID WHERE
APPENDIX F o
E.l
DAMPING IS PURELY HVSTERETIC "
,
NUMER 1CAL METHOD - PROGRAM SAMPLE ,,' ::-
F.I
P.l
Punction of Pro gram
F',2
Pl'ogram Stl'uctul'e
F.I
F.3
Descl'iption of Pl'ogl'am
F.2
P.3.1
'
F.I
Difficulties Encountel'ed in the Solution
, F.2
t
- viii
e
(1 /
Page
F.'4
F.S F.6
F.3.2
Prooedure of the SoZution
F. 3.3
De8oription of Subroutine8
F.4 ~
Input ==.....=:::::::::~_
.
.
r
OA~------~------~------~------~--------~~----~
o
•
2.
1.
1
~
__ ,
'r~"
Figure 6.
:
,
"
u
"
,
Regions of inst~llity associated with the flrst mode of an undamped clamped-clamped tUbe for a . ,constant excitation parameter:. y=10, S-O. 2 ~
~,
..
<
,
·e \
l
__--x _----llj"'-j-('
~---J-:
1.6
8 =0.2 o
~
8=0.5
-:....- 8=0.8 1.0
o •
0.8~----~~----~------~------~------~0' 0.4 Q "
Figur~ 7.
Regions of %J;lstabili ty associated with the fïrs,t mode of an undamped clamped-clamped tube showin9. the ef~ect .of u=3 i ' y=10. . .
a;
.
.
/'
,
,
\
,
.
o
,~ >
-===-=:=:==~ _.- U-l -.
~~--;:;;':'~-::::-~- oS _ _ ....
... _ _ -_ _ - _ - -_-: - _ - - ..li.- _-
.-' -
~____~d: ~-_?' :-~:§I~:
:.: :
U=1.8
1.8 o
U-3.
0=0. 0=
1.6
2 x 10-4
.,.4 -'-- o=5X10 -3 -.-Ir·- 0=10 Cl
1.0
-U=l' .".,....__ -..J*
,;-~
",,,
'L.
'-'"
~ ...'.~
- A.l -
...
APPENDIX A THE'CHARACTERISTIC DETERMINANT OF THE PRIMARY REGIONS The eléments of the characteristic determinant whi~h
yields thé primary regions of instability (17), when
,
k=5, are as fol1ows: "
~
=
'J 3
[Y 12]
=
u 2 u CC]
[Y 1
3]
=
,k2U 2 [C]
[Y 1 It]
=
0
[,y 1 5 ]
= - 2!3u\.lO[B]
[Y 1 1 J
[C J
y [G]
+ rFJ
-
p. 0
2
fEJ
) .;)
_
4
50
[y 16 J - -2(arF~ + t5 ['E.J
e·
(Y 21]
= u 2 u(C]
[Y 22]
=
'J 3
(Y 2-3]
=
u 2 'J .. (Cl
[Y 2 .. ]
= - Bu\.lO [B]
[Y 15]
=
[Cl
+ 28u [B) )
+ y [G] + tFJ
- in
2
rEl
~;..
30 - r(arFJ + ô tEl + 2au [B» ,
ft
-
'Il
r
,!
-
A.2
-,
è'IJ
.
r li'
[Y 26
]
=-
2f3u\1S'2[B]
• [Y 3 ~]
. [y 3 2 ]
[y 3
3]
2
--
2
= !.U 11 [cl
4
-
fi
= u 2 v .. [C]
=
VI
[Cl
11- y [G]
[Y 3 .. ]
=-
n 2{CX [r'J
[Y 35]
=-
f3Ul1n [B]
+
Ô
+
~F.J
:
~2 rEJ
['E.J + 2f3u[13])
•
[~] = 0
~
,---,.-r
[y ..
1]
=
[y ..
2 ]
= f3Ul1f2[B]
\
-
,-
0
./
[y ..
3]
=
n ï(cx rFJ + 6 tEJ +
2f3u [B])
'--r ....
"
[y .. Il] =
'~-i
[Cl
= U 2 V S [C]
[y ..
=
6]
, [Y's;l
k4
+ "([G] s+'tFJ
[y .. 5]
[Y 5 1]
"e
V2
2
•
tEJ .
îU~ 11 2 [Cl
"
=. 26Ul1f2 [B] "
.ln
= r{a tFJ
+ 15
tEJ +
"'", '::
'(1
26u[B])
t
.
,
~.
--
C!J
/
J ~
, "-,
"
.;
\,\'
.; 1
.
- A.3 -
, [y 53]
= SUllr2'[B]
[Y5l+]
= U 2 V S [C]
:
[y 5 5]
[Y 61
.
=
:=,
+ ['FJ -.
[C] + y [G]
V 3
SQ
]
.
\
2(0 ~F.J
9
l'? 2 'tEJ
. é '
.
\
0
+ ô rEl + 2Su [B])
[{0
•
.fJ Q
f Y62 ]"-=
2t3u~r2[B]
. (Y63]
' [ Y 61t
]
'If
=0 =
.k2~2 [Cl 4
11
2
[Y 65]
=
U
-tY 66
==
V 3
]
11[C]
[Cl , ... 'Y [G]
+ tFJ
~.
6
- p2 tEJ
It should be noted ,that the first elements in the (
determinants when we take k=+ and k=3 are respectively those / d
corresponding.. to [y 33] and [Y 22] listed above'. '
o
o
(
-t
t·
:~~~ .. I. ~_._
\
- B.l -
APPENDIX B THE CHARACTERISTIC DETERMINANt OF THE SECQNDARY~ REGIONS' "
o The elements of the characteristic deterrninant
.
which yièlds the secondary regions of instability, when k=4, 1
are:
,\
.
,
3
~U110 [B]
[z 21 ] =
J
u 211t" [C]
o
[Z
.
21t
] " . - O(arFJ + ISrEJ + 2Bu{~])
[z 2S ] = -'~U"n[B] 2~ t"
.e
"
~~;,
;.;0 ~_
. "
- (J3.2 -
[Z32]
= }BUllQrB] ')
3]
=
V 3 [Cl
. [z3It]
=
U
J
=
iu
[Z .. 1]
=
~uun
[Z ..
2]
=
n (Cl [TJ + Ô ['EJ + 2Bu [B] )
[Z .. 3]
=
2u 2 11[C]
[Z .... ]
= V 7 [Cl
[Z 3
[Z 3
5
r,
2
+ y[G] +
rFJ
11[C]
2
11
2
3
[Cl
[B]
~
+ y [G] + [FJ - n 2 rEl
2 [z .. s] = U 11[C]
[Z 5 l ] = 2n (Cl ~FJ + ô tEl + 2Bu[B])
[Z 5 2.]
= îaUlln [B]
[Zp]
= }x2112 [C1 4
[Zs .. ]
= u 2 u[C]
[Zss]
= v 3 [Cl
+ y fG] +
tF.J
-
4$12
tEJ
- B.3 ... c
It should be noted that the ~irst element in the determinant when k=2 corresponds to [Z22] listed above. k=O, q de terminant of one elernent
([Z33])
For
is obtained •
.
.lI
l
/
\
.....
()
(
.
•
'
,
,'"
t - C.I -
APPENDIX C . THE CHARACTERISTIC DETERMINANT OF THE •
PRIMARY
REGIONS (REFINED THEORY) f
The elements of the characteristic determinant
.
which yields the prJmary regions of instability, using the refined theory
[WII]
= v,[C]
[w12 ] =
2 U 1.1
disc~ssed
in Chapter VI, are:
+ y[G] +' tF.J - ~5n2tEl
[C]
[W 11t J
=
.[W15
]
=
[W16
]
:::
[W 21
]
= U 2 li\[C]
0
~uJ,ln (3 (B] +
... ~ il) f"FJ< +
[0])
[Cl
ô [El + 2Bu [B))
1
[W 22
[W 2
]
3]
,
="'v 3
[Cl
+ y [G] + tFJ'
-
~2 tEJ
= u 2 v .. [Cl .
[W 2
.. ]
= - ~Ul1Cl ([B]
[W 2
5]
30 = - r(atFJ
+
[Cl
...
\ [D] )
('
,
+ 6 tEJ +. 2Bu [B) )
p
- C.2 -
~/3u~n~5 [B] +
[w 26]
=
tw 3 l J
2 2 ll [Cl = .!.U 4
[w 32] =
-
[0]
-
[C))
U2 Vlf [Cl ~
[w 33]
[w , 31+]
=
\) 1
=-
[Cl
+
n
Y [G]
ï(a ['FJ +
..,
~
+
['FJ
rEl
+
-
.'
~n2 "["EJ "
2/3u[B]
-
Bu[G])
c
[w 35 J
=
-.
~ /3UlJn (3 [B] +
[0]
- , [C)) c-
[W 36]
=
0
{w .. 1] = 0
t [w.. s ]
=
U 2V 5
[C]
•
..
- C.3 -
e
~
[W 53] =
~su~n ( [B] +
[Cl
-
+
~F.J
[0»
'#
=
U
[W 55
=
\) 3
]
" [W 5 G] =
[W 6
't l' ".
\
U
2
'J 5
[Cl
{Cl
+
Y [G]
).l[CJ
-
~2 tEJ
.
~
(a [T.J + IS ['EJ + 2Bu[B]) = sn 2
1]
=
[W G 2]
1eUl1SH3[B)
+ [Cl
-
1
[D»
"
U'
[W 63 \
2
[W 5 It]
J = 0
[W G .. ]
2 2 [C1 = .k 4 1J
[W G 5]
=
u~}J[C)
=•
\) 3
1
(W 6 6 )
{Cl + y [G]
f' ..
i
+ , ['FJ
- ~2 rEl
--B.l -
,
APPENDIX D
THE CHARACTERISTIC DETERMINANT OF THE SECÔNDARY REGIONS (REFINED THEORY) The elements of the characteristic determinant whidh yields the secondary regions of instability, using the refined theory discussed in Chapter VI, are:
[x 1 1]
=
V 3
[e]
+
y [G]
+ rFJ - 40 2 rEl
[ X 1 2 1 = u 2 "~ [Cl
[x 1 3] = 0
[X lit]
=
[X 1 5]
=
"
/
~u~n (2 [B] + -
.i.
[0])
20 (a rF,J + ô ["El + 2f3u[BJ)
[X 2
1]
= u 2 ~dC]
[X 2
2]
=
[X 2
3]
= -
(X 2
.. ]
=
V6
[Cl
[Cl + y [G] + ['FJ
Bu~n
( [Cl
-
- n 2 ['E.J
[D])
- n (a ['FJ + ~ tEl + 2t3u
.
[X 2S ]
=
[X, 1)
=0
~aU~O(4{~] +
.(0)
:'"
[Cl)
•
/
- 0.2 -
.
\.J
[X 3 2]
=.~SUlln(2[B] +
[0]
-
[e])
C' [X 33]
=
[X 3 .. ]
= U2 11 [Cl
Ix 35 ] = [X ..
1]
\) 3
k4
[Cl
2
+ y
[G]
+ [Tl
.
11 2 [Cl
= îBu~n (4 [B]
[X" 2]
= n (CL tF,J
[XIf3]
=
lX .. It]
=
+ [0]
-
[C]) u
+ Ô [E] + 2Bu[B])
2u 211[C]
"\)?
[Cl :+ y [G] +
tFJ - n 2 ['E.J
2 [Xit 5] = U 11[C]
«l ['F.J
,""
+ ô rEl + 2au[B])
[X 5 1 ]
= 2
[X S 2]
= ~Ulln (2 [B]
+ [Cl
-
[0] )
,k2 11 2[Cl
[X S 3] = 2 (X SIt
]
=
[X S 5] =
U2~[C]
\) 3
[C1 + y [G] +
tFJ
- 4n~
tEl
r"
\
, ." / t
{
"
,
\r
- E.l
Af'PENDIX (-
\
~
ANALYSI~
1
EOR,,[\ DAMPED TUBE CoNVEYING ELUID WHERE .
- 1
In
presented the analysis performed ,
to incorporate the damping.
f ct of the purely hysteretic internaI
This effect is represented mathematically by replacing
the internaI damping term in the equation of motion (4) by El (v!wf) (a S y/ax 4 at) where
~
of~ysteretic
is the coefficient
damping and ia independent of the circular frequency of the steady-state motion, wf, provided v«l (Bishop & Johnson (1960). Then, we proceed as in
§
3.2 dntil we obtain the new equation
of motion in its matrix forro (1)
Following Bolotin's method, we seek a solution in the forro
.,., ". (II)
to determine the primary regions'of instability. From equation (II) we ean see that the motion of the tube may be expressed as a summation of simple harmonie motions of .
frequ~ncies
kG/2.
Therefore, (II~)
- E.2 -
...
Substituting '(II) and (III) Il1nto (I)' and proceeding as in
.
.§
charaet~ristic
4.2, we obtain .the new
determinant where
the on1y change, as compared to the old one (Appendix A) , appears in the terms that include the coefficient of internaI r
.
dubstitute (II) and (III) (2v rFJ )(kfl .,
"
k
2
= v ~FJ
!E
00
k=1,3,5
/
Ë k=1,3,5
{-a} k
From
!
•
~nto
(1), these terms k~h
cos'~
2
be~ome
... )
{a k } cos ..• ,) .
/ /
"
They wl~} always take the forrn'pfFJ, sinee when we
damping.
~quation
independent of k and
d"
(IV)
\ (IV) hysteretic damping is shown to be Aocordingly (beeause of the non-
dependence on k) the eharacteristic determinant
obt~ined
previ-
ous1y to determine the secondary instability regions is changed in the same way as deseribed above for the primary regions; i.e .. the hysteretic damping term in the determinant will also have the forro
'.
vrF~. ,
""'.
This analysis of the system with hysteretic dissipation
.
is quite simplistic, as may be verified by consulting' for'
.
\
instance Mozer & Evan-Iwanowski (1972) paper on the stability of a hysteretically damped column subjected to an oscillatory •
load.
.
.
"'r'
..
•
.. ",
,
,
- F.l ...
APPENDIX E· NUMER 1CAL METHOD
P.l
,~
PROGRAM SAMPLE
,-"""
Punation, of Program The:program,dete~ines
the regipns of instability
for a tube'conveying pulsating.flow.
,
It is written in a
.'
general forro so that tUbes'with different boundarYiconditions f
(e.g. clamped-clamped, clamped-free, etc . . . . ) "
m'y
be studied.
Tnis is done by replacing only one or two subroutines, depending on the case under consideration, by those corresponding te the l,
new system~ F.2
l
Program Struature
The whole program i5 written in Fortran IV language; hence, standard control
~ards
are' _vseii.
It can be run on the
I.B.M_ 360/75 digital computer. The program has the following structure:
1
Main .-'
SUBROUTINE ROUGH )
(!t.
'-,
SUBROUTINE ACURAT SUBROUTINE DETMAT
SUBROUTINE EICOEFF SUBROUTINE OETER AlI calculations are carried out with double precision.
, ~{ ! ','
~''''''' .,.-... ...... OCOOOo
'-
,
--. .
c •
:: * .....