San Diego, CA 92103Multimode Fluid Elastic [628154]

R. D. Blevins
Goodrich
San Diego, CA 92103Multimode Fluid Elastic
Instability of Heat
Exchanger Tubes
Multimode fluid elastic instability analysis is made of heat exchanger tubes in cross flow.
The stability analysis predicts that the flow velocity for onset of tube instability in nonuni-
form flow is lowered by participation of multiple tube modes with similar natural frequen-cies. [DOI: 10.1115/1.4025604]
1 Introduction
The vibrational instability of heat exchanger tubes in external
cross flow, called fluid elastic instability, is well known [ 1–6]. It
causes large amplitude destructive tube vibrations when cross
flow over tubes exceeds a critical velocity. However, the literatureonly has single mode analysis, which is appropriate when flow is
uniform or natural modes of the tubes are well separated in fre-
quency. This study considers nonuniform over the tube and the
natural frequencies of the tube modes are not well separated, as is
appropriate for multispan tubes such as shown in Fig. 1. The sta-
bility analysis uses a coupled displacement fluid model; an alter-
nate approach with modal superposition theory gives the same
result.
2 Fluid Elastic Theory
Consider cross sections of three adjacent tubes in cross flow
shown in Fig. 2[3]. The tubes are numbered j/C01,j, and jț1.
They have mass uniform mper unit span, including added mass,
and are elastically supported. Tube displacement parallel and per-
pendicular to the mean flow velocity Ubetween tubes are denoted
byxandy, and distance along the tube span zextends from z¼0
toz¼Ln. The tubes are modeled as uniform elastic beams with
modulus of elasticity Eand area moment of inertia I. The linear
partial differential equations of motion of the jtube in the xdirec-
tion and the jț1 tube in the y direction include the fluid coupled
forcing terms on the right-hand sides [ 2,4].
EI@4xj
@z4țm@2xj
@t2¼1
2qU2ðzȚCxyjț1(1a)
EI@4yjț1
@z4țm@2yjț1
@t2¼/C01
2qU2ðzȚKyxj (1b)
The dimensionless fluid coefficients CxandKycouple the xandy
displacements of adjacent tubes, which oscillate 180 deg out of
phase. The fluid density is qandU(z) is the mean cross velocity,
perpendicular to tube axis through the minimum gap between
tubes. U(z) varies along the span as shown in Fig. 1, bottom.
The homogeneous ( Cx¼Ky¼0) solutions to Eqs. (1)with the
appropriate boundary linear conditions at tube supports are the
tube mode shapes ~xj;i;~yjț1;i,i¼1,2,3… Nand natural frequencies
xx,j,k,xy,jț1,kin radians per second, which are integrals of the
mode shapes.x2
xj;i¼EI
mðLn
0ð@4~xj;iðzȚ=@z4Ț~xj;iðzȚdz
ðLn
0~x2
j;iðzȚdz;
x2
yjț1;i¼EI
mðLn
0ð@4~yjț1;iðzȚ=@z4Ț~yjț1;iðzȚdz
ðLn
0~y2
jț1;iðzȚdz(2)
The mode shapes are orthogonal over the span of the tubes
ðLn
0~xj;iðzȚ~xj;kðzȚdz¼ðLn
0~yj;iðzȚ~yj;kðzȚdz¼0;j6¼k (3)
see Appendix A.
The tube displacements are expanded in a series of their first N
modes
xjðz;tȚ¼XN
i¼1Xj;iðtȚ~xj;iðzȚ;yjț1ðz;tȚ¼XN
i¼1Yjț1;iðtȚ~yjț1;iðzȚ(4)
Substituting Eqs. (4)into Eqs. (1a)and(1b), multiplying through
by the mode shapes ~xj;k;~yj;k, respectively, and using the orthogon-
ality of modes over the total tube length Ln, Eq. (3), gives the lin-
ear coupled modal equations of motion of adjacent tubes
m€Xj;kðtȚț2mfxj;kxxj;k_Xj;kðtȚțmx2
xj;kXðtȚ
¼1
2qCxXN
i¼1Yjț1;iðtȚðLn
0U2ðzȚ~xj;kðzȚ~yjț1;iðzȚdz
ðLn
0~x2
j;kðzȚdz
m€Yjț1;kðtȚț2mfyjț1;kxy;jț1;k_Yjț1;kðtȚțmx2
yjț1;kYðtȚ
¼/C01
2qKyXN
i¼1Xj;iðtȚðLn
0U2ðzȚ~xj;iðzȚ~yjț1;kðzȚdz
ðL
0~y2
j;kðzȚdz(5)
Xand Yare modal displacements. The first subscript is the tube
number and the second is the mode. The overdot denotes deriva-
tive with respect to time. Linear structural modal damping terms
proportional to the dimensionless damping factors fxjandfyjț1
have been added in. The two equations are coupled by the fluid
terms proportional to CxandKy. If the flow velocity is a function
ofz, that is, nonuniform over the tube length (Fig. 1), then, as
noted by Paidoussis [ 1], modal expansion does not uncouple theContributed by the Design Engineering Division of ASME for publication in the
JOURNAL OF VIBRATION AND ACOUSTICS . Manuscript received August 31, 2012; final
manuscript received September 12, 2013; published online November 19, 2013.
Assoc. Editor: Walter Lacarbonara.
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modes. Each mode is fluid dynamically coupled to all N modes of
the adjacent tubes. Here we proceed with the coupled solution.
To simplify the notation, the first index subscript is dropped
with the understanding that xrefers to inline displacement of j
tube and yrefers to transverse displacement of the adjacent jț1
tube. Cross modal flow velocities are defined by akiand bkithat
have units of velocity squared
aki¼ðLn
0U2ðzȚ~xkðzȚ~yiðzȚdz
ðLn
0~x2
kðzȚdz;bki¼ðLn
0U2ðzȚ~xiðzȚ~ykðzȚdz
ðLn
0~y2
kðzȚdz(6)
If the modes are normalized to the same mean square value, then
akiand bkiare symmetric. The 2 Nlinear coupled equations of
motion are
m€XkðtȚț2mfxkxxk_XkðtȚțmx2
xkXkðtȚ¼1
2qCxXN
i¼1akiYiðtȚ;
m€YkðtȚț2mfykxyk_YkðtȚțmx2
ykYkðtȚ¼/C01
2qKyXN
i¼1bkiXiðtȚ;
k¼1;2;3; :::N (7)
Solutions are sought such that the displacements either grow or
decay exponentially in time
XiðtȚ¼ /C22Xiekt;YiðtȚ¼ /C22Yiekt(8)Equations (8)are substituted into Eqs (7). The result is put into
matrix form
n00
0Lxk0
00 n2
6643
775/C22Xk8
>><
>>:9
>>=
>>;¼1
2q
mCx aki2
6643
775/C22Yk8
>><
>>:9
>>=
>>;
n00
0L
yk0
00 n2
6643
775/C22Y
k8
>><
>>:9
>>=
>>;¼/C0 1
2q
mKy bki2
6643
775/C22X
k8
>><
>>:9
>>=
>>;(9)
The diagonal matrices on the left contain the tube mass, stiffness,
and damping
L
xk¼k2ț2xxkfxkkțx2
xk;Lyk¼k2ț2fykxykkțx2
yk(10)
The entries of the N/C2Nsymmetric modal velocity matrices [ aki]
and [ bki] on the right are given by Eq. (6). Since [ Ly] is a diagonal
matrix, the second line of Eq. (9)is easily solved for the displace-
ment vector { Y} in terms of { X}. The result is substituted into the
first line of Eq. (9)to create a N/C2Nlinear system of equations
½LD/C138f/C22Xg¼0;LD ii¼1ț1
4q2
m2CxKy
LxiXN
k¼1aikbki
Lyk;
LD ij¼1
4q2
m2CxKy
LxjXN
k¼1aikbkj
Lyk;i6¼j (11)
For solutions other than the zero solution, the determinant of the
[LD] matrix must be zero [ 7]. Setting the determinant of [ LD]t o
zero results in a stability polynomial in k. For stability, kmust
have only negative real parts so perturbations die out in time, Eq.
(8)[7]. The critical velocity for onset of instability is the flow ve-
locity at which kpasses through zero and becomes positive.
3 Solutions for Critical Velocity
3.1 Single Uncoupled Mode, N51.A check is made against
existing single mode stability analysis [ 2,4]. For i¼j¼N¼1a
fourth order stability polynomial is formed by setting the single
term [ LD] matrix to zero.
ðk2ț2fxxxkțx2
xȚðk2ț2fyxykțx2
yȚț1
4q2
m2CxKya11b11¼0
(12)
If the xandymodes have the same mode shape a11¼b11, which
is the case if tubes are straight, this stability polynomial identicalto Eqs. (5)–(18) of Ref. [ 2] and stability analysis [ 1–3] produces
Connors’ equation for onset of fluid elastic tube instability of a
single mode
Ue
fnD¼Cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
mð2pfȚ
qD2s
(13)
C¼2(2p)1/2/(CxKy)1/4. The modal effective velocity is the square
root of a11[2], as is discussed in Appendix B.
U2
e¼a11¼ðLn
0U2ðzȚ~x2ðzȚdz
ðLn
0~x2ðzȚdz(14)
Fig. 1 Three cases of multiple span heat exchanger tubes with
nonuniform flow
Fig. 2 Elastically supported tubes in cross flow [ 2]
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Instability, that is destructive large amplitude vibration, commen-
ces when the effective flow velocity Eq. (14), reaches the critical
velocity Eq. (13).
3.2 Two Coupled Modes. Consider that two modes of each
tube participate in the instability, N¼2. Setting the determinant
of the two by two [ LD] matrix (Eq. (11)) to zero, produces an
eighth order stability polynomial.
Lx1Lx2Ly1Ly2țțðq2CxKy=4m2Ț2ða12a21/C0a11a22Țðb12b21/C0b11b22Ț
țðq2CxKy=4m2Țða11b11Lx2Ly2ța12b21Lx2Ly1ța21b12Lx1Ly2
ța22b22Lx1Ly1Ț¼0 (15)
First consider the case where there is no modal coupling,
a12¼a21¼b12¼b21¼0 such as for uniform flow. In this case,
Eq.(15) uncouples to the single mode solution Eq. (12) for eachof the two modes. Similarly, if a mode has a much greater natural
frequency (stiffness) than the second mode, then the stability also
reduces to the single mode analysis.
Now consider the case where the natural frequencies of two
coupled modes are closely spaced. This case is justified by both
the very close frequency spacing of multispan tubes (Appendix A)
and by experimental results that show the instability is relatively
insensitive to small frequency differences between tubes [ 8,9]. If
Lx1¼Lx2¼Ly1¼Ly2, Eq. (15) can be factored into two quadratic
terms that are set to zero. The solution is
L2
x1¼/C01
4q2
m2CxKyU4
e (16)
This equation is identical to the single mode instability
polynomial (Eq. (12)) but with a redefinition of the effective ve-
locity Ue:
U4
e¼ð1=2Țða11b11ța12b21ța21b12ța22b22/C7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ða11b11ța12b21ța21b12ța22b22Ț2/C04ða12a21/C0a11a22Țðb12b21/C0b11b22Țq /C19
(17)
The Eq. (13)stability criterion applies but with this definition of effective velocity.
If modes in the xandydirections are identical then aij¼bij, as is appropriate for straight tubes, athe effective velocity (Eq. (17)) for
two-mode coupled analysis is
U4
e¼1
2a2
11ța2
22ț2a12a216ða11ța22Țffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ða11/C0a22Ț2ț4a12a21q /C20/C21
ðaȚ
a2
11;or;a2
22; fora12¼a21¼0ðno coupling ;Eq:ð14ȚȚ ð bȚ
a116ffiffiffiffiffiffiffiffiffiffiffiffi ffia12a21p/C0/C12;for a 11¼a22 ðcȚ8
>>>><
>>>>:(18)
Fora
11¼a22the square of effective velocity increases from a11
for the uncoupled single mode analysis to a11ț(a12a21)1/2for
coupled modes. Written out this is
U2
e¼ðLn
0U2ðzȚ~x2
1ðzȚdz
ðLn
0~x2
1ðzȚdzțðLn
0U2ðzȚ~x1ðzȚ~x2ðzȚdz/C12/C12/C12/C12/C12/C12/C12/C12
ð
Ln
0~x2
1ðzȚdz/C20/C21 1=2ÐLn
0~x2
2ðzȚdzhi1=2(19)
The first term is the single mode effective velocity, Eq. (14).
The second term is modal coupling contribution. For nonuniform
flow, it increases the effective velocity and, thus, lowers the
physical cross flow velocity for onset of instability of coupled
modes.
3.3 Two Coupled Modes by Modal Superposition. Modal
superposition provides an alternate avenue for coupled mode sta-bility analysis that does not require fluid displacement theory.
Consider the tube response is the in-phase supposition of two or
more natural modes with closely spaced natural frequencies
~xðzȚ¼ ~yðzȚ¼ A ~y1ðzȚ
ðLn
0~y2
1ðzȚdz/C20/C21 1=2ț~y2ðzȚ
ðLn
0~y2
2ðzȚdz/C20/C21 1=2(20)
The mode shapes are nondimensionalized for convenience of
notation. The scalar proportionality Adetermines the relative con-
tribution of each mode to the overall response.Single mode analysis is applied with this supermode. The effec-
tive velocity of the two-mode supermode is calculated by substi-
tuting Eq. (20) into Eq. (14). Using the orthogonality, Eq. (3), the
result is
U2
e¼A2a11ț2Aða12a21Ț1=2Signða12Țța22/C16/C17
=ðA2ț1Ț(21)
Sign( a12) is the sign of a12, Eq. (6). The derivative of Ue2with
respect to Ais set to zero to find the value of Athat maximizes
effective velocity
A¼6ða11/C0a22Ț6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ða11/C0a22Ț2ț4a12a21q /C18/C19
=2ffiffiffiffiffiffiffiffiffiffiffiffi ffia12a21pðȚ
(22)
Substituting this into Eq. (21) gives the maximum and minimum
effective velocities for two coupled modes
U2
e/C02mode¼1
2a11ța226ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ða11/C0a22Ț2ț4a12a21q /C20/C21
(23)
When squared, this gives the same result as displacement fluid
force theory, Eq. (18a). Both expressions reduce to the single
mode expression (Eq. (14)) when the velocity is uniform over the
entire span (Fig. 1, middle).
3.4NCoupled Modes. Now consider Ncoupled modes. The
solution to Eq. (11) for critical velocity is simplified by assuming
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that each mode contributes equally, X1¼X2¼…XN,aij¼bij¼a11
ifi¼janda12ifi6¼j, and Lxi¼Lyj¼Lx1for all iandj. Each row
Eq.(11)then becomes
L2
x1¼/C01
4q2
m2CxKyða11țðN/C01Ța12Ț2(24)
which corresponds to a critical velocity that increases with the
number of cross terms, that is, the number of modes
U2
e/C0Nmode¼a11țNa12 (25)
This is consistent with Eqs. (18) and(23) forN¼2 and it illus-
trates the increasing importance of modal coupling to critical ve-
locity as the number of closed passed modes increases.
4 Example
Consider the eight span tube with clamped ends and pinned in-
termediate spans shown in Fig. 1, top. The spans are equal length.There is uniform velocity Uoon the first span and no flow on the
remaining spans. Their mode shapes are computed in Appendix A
and shown in Fig. 3. Table 1contains the modal velocity matrix,
Eq.(6), for these modes. Diagonal terms are the square of the
effective velocity, Eq. (14), for each uncoupled mode. The lowest
velocity Uofor onset of instability of a single mode is when the
product fnUe/C24kaii1/2is minimum and this is the case for mode 7.
The single mode effective velocity for mode 7 is Ue¼a771/2
¼(0.2364)1/2Uo¼0.482 Uo, about one-half the physical velocity
Uoover the first span.
The coupled mode instability critical velocity (Eq. (19))) of the
two adjacent mode pairs is in Table 2. The most critical mode pair
is the coupled instability of the sixth and seventh modes and this
produces a coupled instability at the effective velocity Ue¼0.665
Uo, which is a factor of 1.37 higher than mode 7 alone; thus, it
predicts an onset of instability at a flow velocity Uothat is a factor
36% lower than the uncoupled stability analysis of mode 7.
Single mode analysis gives an effective velocity of 0.482 Uo.
Coupled mode analysis increases this to 0.665 Uo. The most con-
servative estimate of effective velocity is 1.0 Uo, which is
obtained by considering uniform flow over the entire tube length
or by making analysis span by span.
The mode participation is computed from Eq. (11) at onset of
instability
~X2
~X1¼a22/C0a11țffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ða11/C0a22Ț2ț4a12a21q
2a12(26)
In the example this works out to X7/X6¼1.07. The combination
of the sixth mode plus 1.07 times the seventh mode in Fig. 4cre-
ates a supermode that has a higher effective velocity than either
mode by itself.
5 Discussion
The analysis suggests that multiple tube modes with similar nat-
ural frequency can couple dynamically during fluid elastic insta-
bility. Experiments with multispan tubes generally support this.
Goyder [ 5] found that “vibrating neighboring tubes with may cou-
ple with the tube being assessed to enhance the fluid elastic insta-
bility.” Weaver and Parrondo [ 10] found that multiple modes
were excited during rise to instability of multispan tubes. The
modal coupling discussed in this paper, is similar to the added
mass coupling, demonstrated by SS Chen [ 11], that does not input
energy but creates coupled tube modes similar to those considered
here.
The example suggests that the reduction in critical velocity is
about 36% for multiple span tubes with nonuniform flow but this
may be difficult to verify from operational plant experience owingto the scatter in reports of velocity for onset of tube instability
[3,5,6,8,9]. Evidence of multimode interaction could be obtained
by wear patterns. In the example, the mode 7 is the single mode
most prone to instability. It has highest response at its first and
last spans so tube wear in seventh mode instability is expected atfirst and last spans. But the sixth țseventh coupled mode’s first
span, the span exposed to high velocity flow, has the highestTable 1 Cross modal velocity matrix aij/Uo2for eight tube span
example
i= 1 2345678
0.0104 0.0201 0.0288 /C00.0358 /C00.0412 /C00.0449 /C00.0472 /C00.0340
0.0201 0.0391 0.0559 /C00.0697 /C00.0802 /C00.0875 /C00.0922 /C00.0664
0.0288 0.0559 0.0799 /C00.0998 /C00.1151 /C00.1260 /C00.1330 /C00.0959
/C00.0358 /C00.0697 /C00.0998 0.1250 0.1446 0.1588 0.1682 0.1215
/C00.0412 /C00.0802 /C00.1151 0.1446 0.1679 0.1854 0.1972 0.1427
/C00.0449 /C00.0875 /C00.1260 0.1588 0.1854 0.2057 0.2200 0.1596
/C00.0472 /C00.0922 /C00.1330 0.1682 0.1972 0.2200 0.2364 0.1718
/C00.0340 /C00.0664 /C00.0959 0.1215 0.1427 0.1596 0.1718 0.1250
Fig. 3 Mode shapes of an eight span beam with clamped ends
Table 2 Modal effective velocity for single and multimode anal-
ysis, Eqs. (14) and 19
mode,
i ki aii/Uo2aij/Uo2
j¼i/C01a ii1/2kiSingle
Eq.(14)
Ue/UoMultiple
Eq.(18)
Ue/Uo
1 3.2101 0.0104 0.1825 0.1019
2 3.3932 0.0391 0.0201 0.3643 0.1978 0.2223 3.6454 0.0799 0.0559 0.5398 0.2827 0.3454 3.9266 0.1250 /C00.099 0.7006 0.3536 0.453
5 4.2081 0.1679 0.1446 0.8407 0.4098 0.541
6 4.4633 0.2057 0.1854 0.9582 0.4536 0.6117* 4.6552 0.2364 0.2200 1.0489 0.4862 0.665
8 4.7300 0.1250 0.1718 0.7689 0.3536 0.601
*most critical mode
Fig. 4 Superposition of modes 6 and 7 efficiently responds to
flow velocity over the first span
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response, and so for coupled mode instability the wear would be
greatest on the first span.
The objective of design is to ensure the heat exchanger tubes
are stable across the range of operating velocities [ 3]. For uniform
flow, the means ensuring the physical effective velocities, Eq.
(14), do not exceed the critical velocities of Eq. (13) for any
mode. However, for nonuniform flow and multispan tubes, Eq.
(14), with closely spaced natural frequencies, Eq. (14) is a lower
bound for effective velocity. An improved (higher and, therefore,
more conservative) estimate is to include the coupled mode con-
tribution, Eq. (19). An alternative approach is to superimpose two
modes with similar natural frequency, Eq. (20), and numerically
search for the scalar modal contribution factors that maximize Ue.
An upper bound, and, therefore, most conservative, estimate for
effective velocity is obtained by considering the cross flow to be
uniform over the entire span, or alternately, by making stability
analysis of each span separately.
6 Conclusions
Multimode fluid elastic instability analysis shows that two
coupled tube modes with closely spaced natural frequencies respond
more efficiently to the nonuniform cross flow than a single tube
mode. The result is a decrease in the flow velocity for onset of insta-
bility from that predicted by the single mode analysis for nonuniform
flow over the tube span. An example suggests that the decrease inpredicted critical velocity for onset of instability is on order of 36%.
Nomenclature
aik,bik¼effective modal squared flow velocities, Eq. (6)
C, C x,Ky¼dimensionless fluid force ccoefficients
E¼tube modulus of elasticity
I¼tube moment of inertial
m¼tube mass per unit length including added mass
L, L n¼single and multiple tube span lengths
Lx,k,Ly,k¼linear differential equation operator resultant, Eq. 10
t¼times
Ue¼effective velocity, weighted by mode shapes of one or
more modes
U(z)¼cross flow velocity, a function of z
X¼tube displacement parallel to the mean flow over tubes
Y¼tube displacement perpendicular to the mean flow
over tube
z¼distance along tube span
f¼damping factor, dimensionless
k¼exponential stability parameter, units of t/time; dimen-
sionless frequency parameter
q¼fluid density
x¼2pf¼circular natural frequency, radians per second
Subscripts
i¼tube mode number, 1,2,3…
j¼tube number, 1,2,3…
x¼parallel to flow
y¼perpendicular to flow
Appendix A: Natural Frequencies and Mode Shapes of
Multispan Beams
Consider the straight equal span, multispan beam in Fig. 1. The
two ends are clamped. The Nequal length intermediate spans are
pinned. The natural frequencies have exact solutions and some
unique properties
fi¼k2
i
2pL2ffiffiffiffiffi
EI
mr
;Hz
The dimensionless natural frequency parameter kis given by the
solutions of the following transcendental equation [ 12]:cosmp
N¼1=tanhk/C01=tank
1=sinhk/C01=sink;
N¼number of spans
m¼0;1;2;…N/C01;modes in first pass band and odd pass bands
¼1;2;3:::N;modes in even numbered pass bands
Groups of modes are in frequency pass bands. Between these
bands there are no natural frequencies. The pass bands fall in thefollowing sequences:
m¼1 (first pass band): p/C20k/C204.7308
m¼2 (second pass band): 2 p/C20k/C207.853
m¼3 (third pass band): 3 p/C20k/C2011.0
m¼n(nth pass band): np/C20k/C20/C24p(nț1/2)
In particular, there are Nmodes for an Nspan bean with values
ofkbetween pand 4.7308 and no natural frequencies between
4.7308 and 2 por between 7.853 and 3
p. The modes become more
densely packed as the number of spans increase. For example, foran eight span beam the first eight modes are given by the follow-
ing values of k: 3.210, 3.393, 3.6454, 3.9266, 4.2080, 4.4633,
4.6552, and 4.73004. These fall between 3.14159 and 4.7308. The
next mode, the ninth mode, is k¼6.2832 /C252p.
The mode shapes are computed span by span once the eigenval-
ues have been obtained from the two previous equations. Miles
[13] gives a general expression for mode shape of a single-span
beam with zero displacement at both ends, x¼0 and x¼L.
~y
nðnȚ¼Ansinhksinkn/C0sinksinhkn ðȚ
țBnsinhksinkð1/C0nȚ/C0sinksinhkð1/C0nȚ ðȚ ;
Here n¼x/Landxis distance from the left end. If the left-hand
end of the beam is pinned, then B1¼0. If the left-hand end ( n¼0)
has zero slope, then B1¼{(sinh k-sink)/(sinh kcosk-sinkcoshk)}A1
andA1can be arbitrarily chosen. For continuity, the slope of the
right-hand end of the nth span equals the slope of the left-hand
end of the nț1 span.
Bnț1¼An;n¼1;2;3…N/C01
Anț1¼2Ansinhkcosk/C0sinkcoshk
sinhk/C0sink/C0Bn (A1)
The multispan beam modes are then computed span by span from
left to right. The result is in Fig. 3for an 8-span tube.
Appendix B: Origin of Effective Velocity (Eq. (14))
Equation 14, published in 1977 [ 14], contains the general case
of effective velocity with variable flow velocity over the span. In
the same year Franklin and Soper [ 15] published a particular case
where there is uniform flow over a single span and no flow overthe remaining spans
U
oL2/C0L1
LnðL2
L1~y2ðzȚdz
ðLn
0~y2ðzȚdz
To correct Ref. [ 1], Franklin and Soper [ 15] treat the Fig. 1top
case where as Eq. 14[14] treats the Fig. 1bottom general case
where velocity varies continuously along the span.
References
[1] Paidoussis, M. P., Price, S. J., and de Langre, E., 2011, Fluid-Structure Interac-
tions, Cross Flow-Induced Instabilities , Cambridge University Press, New
York, p. 272.
Journal of Vibration and Acoustics FEBRUARY 2014, Vol. 136 / 011015-5
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[2] Blevins, R. D., 1990, Flow-Induced Vibration , 2nd ed., Krieger, Malabar, FL,
pp. 153–185.
[3] ASME Boiler and Pressure Vesse Code, Section 3, Non Mandatory Appendix
N.
[4] Blevins, R. D., 1977, “Fluid Elastic Whirling of a Tube Row,” ASME J. Press.
Vessel Tech. ,96, pp. 263–267.
[5] Goyder, H. G. D., 2003, “An Assessment Method for Unstable Vibration in
Multispan Tube Bundles,” J. Fluid. Struct. ,18, pp. 555–572.
[6] Weaver, D. S., and Fitzpatrick, J. A., 1988, “A Review of Cross-Flow
Induced Vibrations in Heat Exchanger Tube Arrays,” J. Fluid. Struct. ,2, pp.
73–93.
[7] Bellman, R., 1970, Introduction to Matrix Analysis , McGraw-Hill, New York,
p. 253.
[8] Price, S. J., and Paidoussis, M. P. 1989, “The Flow-Induced Response of a Sin-
gle Flexible Cylinder in an In-Line Array of Rigid Cylinders,” J. Fluid Struc. ,3,
pp. 61–82.[9] Price, S. J., 1995, “A Review of Theoretical Models for Fluidelastic Instability
of Cylinder Arrays in Cross-Flow,” J. Fluid. Struct. ,9(5), pp. 463–518.
[10] Weaver, D. S., and Parrondo, J., 1991, “Fluidelastic Instability in Multispan
Heat Exchanger Tube Arrays,” J. Fluid. Struct. ,5(3), pp. 323–338.
[11] Chen, S.-S., 1975, “Vibrations of a Row of Circular Cylinders in a Liquid,” J.
Eng. Ind. ,97(4), p 1212–1218.
[12] Sen Gupta, G., 1970, “Natural Flexural Waves and the Normal Modes of
Periodically-Supported Beams and Plates,” J. Sound Vib. ,13, pp. 89–101.
[13] Miles, J. W., 1956, “Vibration of Beams on Many Supports,” ASCE J. Eng.
Mech., EM1 , pp. 1–9.
[14] Blevins, R. D., 1977, Flow-Induced Vibration , 1st ed., van Nostrand Reinhold,
New York, Eq. (5)–(8), p. 95.
[15] Franklin, R. E., and Soper, B. M. H., 1977, “An Investigation of Fluidelastic
Instability in Tube Banks Subjected to Fluid Cross Flow,” Proceedings 4thInternational Conference on Structural Mechanics in Reactor Technology(SMIRT), San Francisco, CA, August 15–19, Paper No. 6/7.
011015-6 / Vol. 136, FEBRUARY 2014 Transactions of the ASME
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