transverse vibrations of tensioned pipes conveying fluid with time-dependent velocity

Journal of Sound and <ibration (2000) 236(2), 259}276doi:10.1006/jsvi.2000.2985, available online at http://www.idealibrary.com on

TRANSVERSE VIBRATIONS OF TENSIONED PIPESCONVEYING FLUID WITH TIME-DEPENDENT VELOCITY

HALI0 L RIDVAN OG Z AND HAKAN BOYACI

Department of Mechanical Engineering, Celal Bayar ;niversity, 45140, Muradiye, Manisa, ¹urkey

(Received 26 July 1999, and in ,nal form 12 January 2000)

In this study, the transverse vibrations of highly tensioned pipes with vanishing #exuralsti!ness and conveying #uid with time-dependent velocity are investigated. Two di!erentcases, the pipes with "xed}"xed end and "xed}sliding end conditions are considered. Thetime-dependent velocity is assumed to be a harmonic function about a mean velocity. Thesesystems experience a Coriolis acceleration component which renders such systemsgyroscopic. The equation of motion is derived using Hamilton's principle and solvedanalytically by direct application of the method of multiple scales (a perturbation technique).The natural frequencies are found. Increasing the ratio of #uid mass to the total mass perunit length increases the natural frequencies. The principal parametric resonance cases areinvestigated in detail. Stability boundaries are determined analytically. It is found thatinstabilities occur when the frequency of velocity #uctuations is close to two times thenatural frequency of the constant velocity system. When the velocity #uctuation frequency isclose to zero, no instabilities are detected up to the "rst order of perturbation. Numericalresults are presented for the "rst two modes.

( 2000 Academic Press

1. INTRODUCTION

Due to their technological importance, the dynamics of axially moving continua wasinvestigated by many researchers. The understanding of the vibrations of an axially movingcontinuous medium is important in the design of high-speed magnetic tapes, band-saws,power transmission chains and belts, textile and composite "bers, aerial cable tramways,paper sheets during processing, pipes and beams conveying #uid, etc. Especially, pipe linesare used in conveying gas, oil, water, dangerous liquids in chemical plants, cooling water innuclear power plants, and in many other places. The vibrations occur due to many reasons.They are caused by compressors, ventilators, electrical power engines or internalcombustion engines and also by valves, elbows, ori"ces, transition parts and abrupt areacontractions. If these oscillations are not prevented, they can result in leakage, hazards andaccidents. Ulsoy et al. [1] and Wickert and Mote [2] reviewed the relevant work up to 1978and 1988 respectively. Older studies concentrated on the axial velocity-dependent naturalfrequency and existence of divergence instability at the critical velocity [3}6]. The naturalfrequencies decrease with increasing transport speed, and the translating continuaexperience divergence instability at a critical speed. The eigenfunctions are complex andspeed-dependent due to a convective acceleration component in the equations of motion.The phases of the natural oscillations are not constant and propagate upstream at the phasepropagation velocity. The periodic variation of total mechanical energy was studied [7],and also "rstly, Miranker [8] derived the equations of motion for time-dependent axialvelocity by using variational procedure. Wickert and Mote [9] showed that the energy #uxat a "xed support is the product of the string tension and the convective component of

0022-460X/00/370259#18 $35.00/0 ( 2000 Academic Press

260 H. R. OG Z AND H. BOYACI

a velocity. They [10] also investigated the transverse vibrations by complex modal analysis.An analogous case with a similar equation of motion exists in highly tensioned pipesconveying #uid with negligible #exural sti!ness and #uid pressure e!ect. Wickert [11]analyzed free non-linear vibration of an axially moving, elastic, tensioned beam oversub- and supercritical speed ranges. Pakdemirli et al. [12] re-derived the equations ofmotion for an axially accelerating strip using Hamilton's principle and numericallyinvestigated the stability of the response using Floquet theory. Pakdemirli and Batan [13]analyzed the constant acceleration-type motion. Pakdemirli and Ulsoy [14] obtainedapproximate analytical solutions by using the method of multiple scales and showed thatdirect-perturbation yields better results for higher order expansions with respect to thediscretization-perturbation method. The direct-perturbation method does not requiretransformation of the equations or the selection of an orthogonal basis. The authors alsoinvestigated principal parametric resonances and combination resonances for any twomodes for an axially accelerating strip. OG z et al. [15] studied the transition behaviour fromstring to beam for an accelerating material and presented an approximate analyticalexpression for the natural frequency and determined stability borders for variable velocityand studied principal parametric resonance case. OG z and Pakdemirli [16] investigated thetransverse vibrations of an axially accelerating beam with simple supports. They consideredsix di!erent #exural sti!ness coe$cients and solved the equations of motions by using themethod of multiple scales and studied principal parametric resonances and combinationresonances. They found that for velocity #uctuation frequency nearly twice any naturalfrequency, an instability region occurs whereas for the frequencies close to zero, noinstabilities were detected. For combination resonances, instabilities occurred only for thoseof additive type. No instabilities were detected for di!erence-type combination resonancesin agreement with references [14}16]. OG z et al. [17, 18] investigated linear and non-linearvibrations and performed a stability analysis.

Up to now in the studies related to #uid #ow, the velocity was assumed as constant.Benjamin [19] neglected #uid friction e!ects. Nemat-Nasser et al. [20] and Gregory andPaidoussis [21] found the destabilizing e!ect of dissipation in a cantilevered, fourth orderbeam conveying #uids. Paidoussis and Li [22] and Lee and Mote [23, 24] neglected gravityand pressure e!ects. In references [23, 24] the energetics of translating one-dimensionaluniform strings, highly tensioned pipes with vanishing bending sti!ness and tensionedbeams and #owing #uid are analyzed for "xed, free and damped boundary conditions. The#uid velocity was assumed to be constant. Natural frequencies are found by usingphase-closure principle. It was found that the energies transferred at the di!erent boundarysupports were quanti"ed by energy re#ection coe$cients which were determinedcompletely by the boundary conditions.

In this study, the transverse vibrations of highly tensioned pipes conveying #uid areinvestigated. The pipe is assumed to have negligible #exural sti!ness and it can be thoughtof as a string conveying #uid. A harmonically varying velocity function is chosen for the#uid #ow. The ends of the pipe are on "xed supports in the "rst case and on "xed}slidingsupports in the second case. The linear equations of motion are derived by using Hamilton'sprinciple and solved analytically by means of direct application of the method of multiplescales. The natural frequencies are found analytically depending on #uid velocity and ratioof #uid mass to total mass per unit length and are calculated for the "rst two modes.Increasing the ratio of #uid mass to the total mass per unit length increases the naturalfrequencies. The natural frequencies for "xed}sliding tensioned pipe are lower than those of"xed}"xed tensioned pipe. The stability boundaries are determined analytically forprincipal parametric resonances. For velocity #uctuation frequency nearly twice anynatural frequency, an instability region occurs whereas for frequencies close to zero, no

TENSIONED PIPES CONVEYING FLUID 261

instabilities are detected up to the "rst order of approximation. The stability borders shift tohigher velocity #uctuation frequency values for the two cases. Stable}unstable regions shiftto relatively lower velocity #uctuation frequency values for "xed}sliding end conditionscompared with "xed}"xed end conditions.

2. EQUATIONS OF MOTION

For the "xed}"xed and "xed}sliding tensioned pipes conveying #uid in Figures 1(a) and1(b), x* and z* are the spatial co-ordinates, u* and w* are longitudinal and transversedisplacements respectively. t* is variable #uid velocity, o

fis the #uid density, A

fand A

pare

the cross-sectional areas of the pipe and #ow and assumed to be constant. Tension force inthe pipe is P(t*). The length is ¸. The modulus of elasticity of the pipe is E

p. The transverse

displacement is assumed to be small compared with span ¸ and the tension force is assumedto be su$ciently large compared with the e!ects arising from elongation. The extensionalsti!ness is su$ciently large so that the longitudinal deformation resulting from thepretension is negligible. Variation of cross-sectional dimensions during vibration is notconsidered. In this study sub-critical region is considered. If gravity, pressure and #uidfriction e!ects, and restoring #exural forces are neglected, then a pipe conveying #uid isconsidered to be a string conveying #uid. Let us denote the time by t*, the derivatives withrespect to the spatial variable by ( )@ and the derivatives with respect to time by ( > ). Sinceonly linear analysis is made the non-linear strains deriving from u* and w* are negligible. Sothe strain in the pipe is approximately

e+u*{, (1)

The total kinetic energy of the system is

¹"

1

2ofA

f PL

0

[(t*#uR *#u*{t*)2#(wR *#w*{t*)2] dx*#1

2opA

pPL

0

w5 *2dx*, (2)

where the "rst term denotes the kinetic energies of the #uid in the x* and z* directions, thelast term is the kinetic energy of the pipe in the z* direction. The elastic potential energy ofthe system

;"

1

2EpA

p PL

0

e2dk*#PL

0

Pedx* (3)

relative to equilibrium derives from extension. The "rst term is the elastic potential energyof the pipe due to elongation, and the second term is due to the tensile force. TheLagrangian of the system is

@"¹!;. (4)

The Hamilton's principle is

d Pt*2

t *1

@ dt*"0. (5)

Substituting equation (1) into equation (3), and then substituting equations (2) and (3) intoequation (4), and applying Hamilton's principle (5), the equations of motion for the

Figure 1. Schematics of second order translating continua in a tensioned pipe (a) with "xed}"xed supports, (b)with a sliding end at x"¸.


transverse vibration and the boundary conditions are derived:

wK *#ofA

fofA

f#o

pA

p

(2w5 *{t*#w*{tQ *#w*{{t*2)"0, (6)

w* (0, t*)"w* (¸, t*)"0 (7)

for "xed}"xed tensioned pipe. Following a similar way the boundary conditions for"xed}sliding pipe can be expressed as

w* (0, t*)"w*{ (¸, t*)"0 (8)

The following parameters are introduced to non-dimensionalize equations (6)}(8):

w"

w*

¸

, u"u*

¸

, x"x*

¸

, t"1

¸SP

ofA

f

t* (9)

and the #uid velocity is non-dimensionalized by the critical #ow velocity at which the pipesystem experiences divergence instability

t"t*

JP/ofA

f

. (10)

One obtains non-dimensional linear equations of motion for the transverse vibration

wK#b (2w5 @t#w@tQ #(t2!1)w@@)"0 (11)

with solutions satisfying the end conditions

w (0, t)"w (1, t)"0 (12)


for "xed}"xed tensioned pipe. Following a similar way the boundary conditions for"xed}sliding pipe can be expressed as

w (0, t)"w@ (1, t)"0 (13)

and the ratio of the #owing #uid mass to the total mass per unit length is de"ned as

b"ofA

f(o

fA

f#o

pA

p). (14)

Several cases can be discussed for restricted parameter values. First, for the limiting caseb"1 the equation of motion for a travelling string with variable velocity is obtained asa special case of the second order #uid}pipe system. Second, for a stationary #uid, b"1 andt"0, the equation for linear stationary string is obtained. In equation (11) wK and 2w5 @tdenote local and Coriolis accelerations, respectively, tQ denotes variable #uid velocity, andt2w@@ denotes centrifugal acceleration.

3. APPROXIMATE SOLUTION

Assuming that the velocity is harmonically varying about a constant mean velocity t0,

one writes

t"t0#et

1sinXt, (15)

where e is a small parameter and et-is also small, represents the amplitude of #uctuations.

X is the #uctuation frequency. Substituting equation (15) into equation (11) and keepingterms up to the "rst order of approximation, one has

wK#2bw5 @t0#b (t2

0!1)w@@#eb (t

1X cosXt w@#2t sin Xt w5 @#2t

0t1sinXt w@@)"0. (16)

Direct-perturbation method will be applied to equation (16) in search of solutions. Usingthe method of multiple scales (a perturbation technique) [25, 26] and assuming a "rst orderexpansion, one writes

w (x, t; e)"w0(x,¹

0,¹

1)#ew

1(x,¹

0,¹

1)#2, (17)

where w0

and w1

are the displacement functions at orders 1 and e,¹0"t and ¹

1"et are

the usual fast and slow time scales respectively. Now the time derivatives can be written as

d

dt"D

0#eD

1#2,

d2

dt2"D2

0#2eD

0D

1#2, (18)

where (Di"L/L¹

i). Substituting equations (17) and (18) into equation (16) and separating

each order of approximation, one obtains

O(1) D20w0#2t

0bD

0w@

0!b (1!t2

0)w@@

0"0, (19)

O(e) D20w

1#2bt

0D

0w@

1!b(1!t2

0)w@@

1"

!2D0D

1w

0!2bt

0D

1w@

0!2bt

1sinX¹

0D

0w@0

(20)

!2bt0t1sinX¹

0w@@

0!bt

1X cosX¹

0w@0.


The solution of equation (19) can be written as follows:

w0(x,¹

0,¹

1)"eu

n¹

0>n(x)#cc, (21)

where n is mode number, un

is natural frequency and cc stands for complex conjugate.Substituting equation (21) into equation (19) one obtains

b (1!t20)>@@

n!2bt

0u

n>@

n!u2

n>

n"0. (22)

The solution of equation (22) is

>n(x)"C

1nek

1x#C

2nek

2x , (23)

where k is wave number. Substituting equation (23) into equation (19), one obtainsdispersive relation for the tensioned pipe conveying #uid

b (1!t20)k2

j!2t

0bu

nkj!u2

n"0, j"1, 2. (24)

Solving the polynomial (24) in terms of kjand de"ning k

u"k

1and k

d"!k

2,

kd"

un

bt0#Jb2t2

0#b(1!t2

0), k

u"

un

bt0#Jb2t2

0#b (1!t2

0), (25)

the downstream and upstream wave numbers, respectively, are found. The denominators ofequation (25) are downstream and upstream phase velocities. Substituting equations (23)and (25) into equation (21) and applying "xed}"xed end conditions (12) one obtains

C2n"!C

1n, e!k

d"eku. (26)

In equation (26), !kd"k

uis trivial solution. For non-trivial solution the wave numbers

must be complex:

kd"ikM

d, k

u"ikM

u, (27)

where i denotes complex number J!1, kMdand kM

uare real parts of assumed wave numbers

and the following relationship should hold,

kuM"!kM

d#2nn (28)

to satisfy e!kd"ek

u condition, and of course from equation (25) the assumed naturalfrequency must be complex:

un"iu6

n. (29)

Substituting equations (27) and (29) into equation (25) and then together into equation (28),the natural frequency equation for "xed}"xed tensioned pipe conveying #uid is obtained as

u6n"

nnb (1!t20)

Jb2t20#b (1!t2

0), n"1, 2, 3,2 . (30)


Following a similar derivation for "xed}sliding tensioned pipe conveying #uid, therelationship between wave numbers is obtained as

kMu"!kM

d#2An!

1

2Bn, n"1, 2, 3,2 (31)

and the natural frequency equation is obtained as

u6n"

(n!12)nb (1!t2

0)

Jb2t20#b (1!t2

0), n"1, 2, 3,2. (32)

Combining equations (25), (27), (29), and (30) for "xed}"xed tensioned pipe and equation(32) for "xed}sliding tensioned pipe and substituting into displacement function (16), oneobtains

w0"C

neiu6

n¹

0 (e!ikMddx!eikM

ux )#cc (33)

for "xed}"xed and "xed}sliding tensioned pipes conveying #uid with related naturalfrequencies and wave numbers. Substituting equation (33) into order e equation one obtains

D20w1#2bt

0D

0w@

1!b(1!t2

0)w@@

1"

!2D1C

n(iu6

n>

n#bt

0>@

n) eiuN

n¹

0#2D1CM

n(iu6

n>

n!bt

0>M @

n) e!iuN

n¹

0

!uNnbt

1[C

n>@

n(ei(uN

n#X)¹

0!ei(uNn!X)¹

0!CMn>M @

n(ei(X!uN

n)¹

0!ei(uNn#X)¹

0)] (34)

!ibt0t1[C

n>@@

n(ei(uN

n#X)¹

0!ei(uNn!X)¹

0)#CMn>M @@

n(ei(X!uN

n)¹

0!ei(uNn#X )¹

0)]

!

Xbt1

2[C

n>@

n(ei(uN

n#X)¹

0#ei(uNn!X)¹

0#CMn>M @

n(ei(X!uN

n)¹

0#e!i (un#X)¹

0 )].

In the next section three di!erent cases will be discussed for the "xed}"xed and"xed}sliding tensioned pipes.

4. PRINCIPAL PARAMETRIC RESONANCES

4.1. X IS AWAY FROM 2u6n

AND 0

In this case, for "xed}"xed end conditions, equation (34) becomes

D20w1#2bt

0D

0w@1!b (1!t2

0)w@@

1"!2D

1C

n(iu6

n>

n#bt

0>@

n) eiuN

n¹

0#cc#NS¹, (35)

where cc and NS¹ stand for complex conjugates and non-secular terms respectively. Thesolution of equation (35) is as follows:

w1(x,¹

0,¹

1)"/

n(x,¹

1) eiuN

n¹

0#= (x,¹0,¹

1)#cc. (36)

The "rst term is related to secular terms and the second term is related to non-secular terms.If equation (36) is substituted into equation (35), /

nsatisfy the equations

b (1!t20)/@@

n!2it

0bu6

n/@

n#u6 2

n/

n"2D

1C

n(iuN

n>

n#bt

0>@

n), (37)

/n(0)"0, /

n(1)"0. (38)


The solvability condition requires (see reference [25] for details of calculating solvabilityconditions)

D1C

n"0. (39)

This means a constant amplitude solution up to the "rst order of approximation

Cn"C

0. (40)

Following a similar way one obtains the solvability condition for "xed}sliding pipe as

D1C

n"0. (41)

This means a constant amplitude up to the "rst order of approximation

Cn"C

0. (42)

4.2. X IS CLOSE TO 0

For this case, the nearness of #uctuation frequency to zero can be expressed as

X"ep, (43)

where p is the detuning parameter. Substituting equations (33) and (43) into equation (34),the order e equation is obtained as

D20w1#2t

0bD

0w@1!b (1!t2

0)w@@

1

"M!2D1C

n(iuN

n>

n#bt

0>@

n)!uN

nbt

1C

n>@

n(eip¹

1!e!ip¹1) (44)

#ibt0t1C

n>@@

n(eip¹

1!e!ip¹1)!

Xbt1

2C

n>@

n(eip¹

1#e!ipu¹1 )H

iuNn¹

n

#cc#NS¹.

Following a similar way to that in section 4.1 the solvability condition for "xed}"xed and"xed}sliding tensioned pipes is obtained as

D1C

n#(k

1cosp¹

1#k

2sinp¹

1)C

n"0, (45)

where

k1"

Xbt1 P

1

0

>@n>M

ndx

2Aiu6 n P1

0

>n>M

ndx#bt

0 P1

0

>@n>M

ndxB

,

k2"

u6n P

1

0

>@n>M

ndx#t

0 P1

0

>@@n>M

ndx

iuNn P

1

0

>n>M

ndx#bt

0 P1

0

>@n>M

ndx

i bt1.

(46)


The solution of equation (45) is

Cn"C

0e(!(k

1/p) sinp¹

1#(k

2/p) cosp¹

1) (47)

Since !1)sin p¹1)!1 and !1)cosp¹

1)1, there is no instability up to O(e).

4.3. X IS CLOSE TO 2u6n

The nearness of #uctuation frequency to twice any natural frequency can be expressed as

X"2uNn#ep. (48)

Substituting equations (33) and (43) into equation (34) the order e equation for "xed}"xedand "xed}sliding tensioned pipes conveying #uid is obtained as

D20w1#2t

0bD

0w@1!b (1!t2

0)w@@

1

"G!2D1C

n(iuN

n>

n#bt

0>@

n)

#CMnbt

1eip¹

1AAuN n!X2 B>M @n#it

0>@@

n BH#cc#NS¹, (49)

where CMnis the complex conjugate of amplitude C

n. The solvability condition for this case is

D1C

n#k

0CM

neip¹

1"0, (50)

where

k0"

12(X!2uN

n)P

1

0

>M @n>M

ndx!it

0 P1

0

>M @@n>M

ndx

2Aiu6 nP1

0

>n>M

ndx#bt

0P1

0

>M @n>M

ndxB

bt1. (51)

The solution of equation (50) is assumed as

Cn"B

neip¹

1/2 , CM

n"BM

neip¹

1/2 (52)

Substituting equation (52) into equation (50), one obtains

D1B

n#

ip2

Bn#k

0BMn"0, (53)

where complex amplitudes are

Bn"b

nej¹

1, BMn"bM

nej¹

1 (54)

and bn"bR

n#ibI

n, k

0"kR

0#ikI

0where R and I denote real and imaginary parts

respectively. Substituting equation (54) into equation (53) and separating real and


imaginary parts, one obtains the matrix equation

j#kR0

kI0!

p2

kI0#

p2

j!kR0GbRn

bInH"G

0

0H . (55)

For non-trivial solution, the determinant of the coe$cient matrix must be zero. This impliesthat

j"G

1

2J!p2#4 (kR2

0#kI2

0) . (56)

Stable solution requires j"0. Then the stability boundaries can be written as

p"G2 JkR2

0#kI2

0. (57)

Substituting equation (57) into equation (48) the stability regions can be expressed as

X"2uNnG2eJkR2

0#kI2

0. (58)

The numerical solution for the two pipes will be given in the next section. The solutions di!erin the integrals since the shape functions and natural frequencies are di!erent for the two pipes.

5. NUMERICAL ANALYSIS

In this section, the numerical examples will be given for the vibrations of tensioned pipesconveying #uid. In Figures 2}5, the natural frequencies are plotted depending on mean

Figure 2. The natural frequency value versus mean velocity for di!erent ratios of #uid mass to the total mass(b"0)2, 0)4, 0)6, 0)8, 1)0) for the "rst mode for "xed}"xed end conditions.

Figure 3. The natural frequency value versus mean velocity for di!erent ratios of #uid mass to the total mass(b"0)2, 0)4, 0)6, 0)8, 1)0) for the second mode for "xed}"xed end conditions.

Figure 4. The natural value versus mean velocity for di!erent ratios of #uid mass to the total mass (b"0)2, 0)4,0)6, 0)8, 1)0) for the "rst mode for "xed}sliding end conditions.


velocity and the ratio of #uid mass to the total mass per unit length by using equation (30)for "xed}"xed tensioned pipe and equation (32) for "xed}sliding tensioned pipe for the "rstand second modes. Increasing the mean velocity decreases the natural frequency values. At

Figure 5. The natural frequency value versus mean velocity for di!erent ratios of #uid mass to the total mass(b"0)2, 0)4, 0)6, 0)8, 1)0) for the second mode for "xed}sliding end conditions.

Figure 6. Stable and unstable regions for the principal parametric resonances for the "rst mode (b"0)25) for"xed}"xed end conditions.


the critical speed the frequency values vanish and divergence instability occurs. Increasingthe ratio of #uid mass to the total mass per unit length, increases the natural frequencies. Atthe limiting value, b"1, the results coincide with the results of axially moving string. The


Figure 8. Stable and unstable regions for the principal parametric resonances for the second mode (b"0)25) for"xed}"xed end conditions.


natural frequencies for "xed}sliding tensioned pipe are lower than those of "xed}"xedtensioned pipe. At higher modes the natural frequency values and critical velocity valuesincrease.

Stability analysis is made for the principal parametric resonance case. It is found thatwhen the velocity #uctuation frequency is close to zero, no instabilities are detected up to



the "rst order of perturbation. When the #uctuation frequency is away from zero and twicethe natural frequency, the solutions are bounded and no instability exists. Instabilities occurwhen the frequency of velocity #uctuations is close to two times the natural frequency of theconstant velocity system. The stable and unstable regions are plotted for the principalparametric resonance case by using equation (58) for "xed}"xed tensioned pipe and"xed}sliding tensioned pipe. In Figures 6 and 7, the stable and unstable regions are plottedfor the principal parametric resonance case for di!erent mean velocities and velocity#uctuation amplitudes (b"0)25, 0)75) for the "rst mode and in Figures 8 and 9 for thesecond mode (b"0)25, 0)75) for "xed}"xed end conditions. Increasing velocity #uctuationamplitude enlarges the stability regions. With increasing the mass ratio, the stability shift tohigher X values. At the critical velocity values, the unstable regions widen. In Figures 10 and11, the stable and unstable regions are plotted for the principal parametric resonance casefor di!erent mean velocities and velocity #uctuation amplitudes (b"0)25, 0)75) for the "rstmode and in Figures 12 and 13 for the second mode (b"0)25, 0)75) for "xed}sliding endconditions. The "xed}sliding tensioned pipe shows similar characteristics. Increasingvelocity #uctuation amplitude enlarges the stability regions. With increasing the mass ratio,the stability regions shift to higher X values. At the critical velocity values, the unstableregions widen. The stability borders for "xed}sliding tensioned pipe have lower values thoseof "xed}"xed tensioned pipe. At higher modes the stability borders shift to higher ) valuesfor both cases.

In all "gures (Figures 6}13), the regions in between the planar surfaces are unstablewhereas the remaining regions are stable.

6. CONCLUDING REMARKS

In this study, the transverse vibrations of highly tensioned "xed}"xed and "xed}slidingsupported pipes with vanishing #exural sti!ness and transporting #uid with time-dependent

Figure 10. Stable and unstable regions for the principal parametric resonances for the "rst mode (b"0)25) for"xed}sliding end conditions.

Figure 11. Stable and unstable regions for the principal parametric resonances for the "rst mode (b"0)75) for"xed}sliding end condtions.


velocity are investigated. The sliding end condition for tensioned pipes permits onlytransverse displacement but the slope is zero. The supports result in extension of the pipeduring the vibration and hence introduce further non-linear terms to the equation ofmotion. The #uid velocity is assumed to be harmonically varying about a mean velocity.The equation of motion is solved analytically by direct application of the method of

Figure 12. Stable and unstable regions for the principal parametric resonances for the second mode (b"0)25,)for "xed}sliding end conditions.

Figure 13. Stable and unstable regions for the principal parametric resonances for the second mode (b"0)75)for "xed}sliding end conditions.


multiple scales (a perturbation technique). The natural frequencies are found analyticallydepending on mean velocity and the ratio of #uid mass to the total mass per unit length. It isfound that, increasing the mean velocity decreases the natural frequency values and at thecritical speed the frequency values vanish and divergence instability occurs. Also increasingthe ratio of #uid mass to the total mass per unit length, increases the natural frequencies.


The natural frequencies for "xed}sliding tensioned pipe are lower than those of "xed}"xedtensioned pipe. The principal parametric resonances are investigated in detail. Stabilityboundaries are determined analytically. It is found that instabilities occur when thefrequency of velocity #uctuations is close to two times the natural frequency of the constantvelocity system. When the velocity #uctuation frequency is close to zero, no instabilities aredetected up to the "rst order of perturbation. When the #uctuation frequency is away fromzero and twice the natural frequency, the solutions are bounded and no instability exists.Increasing velocity #uctuation amplitude enlarges the stability regions. With increasing themass ratio, the stability regions shift to higher velocity #uctuation frequency values. Stableand unstable region shift to lower velocity #uctuation frequency values. Stable and unstableregions shift to lower velocity #uctuation frequency for "xed}sliding supports whencompared with those of "xed}"xed supports.

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transverse vibrations of tensioned pipes conveying fluid with time-dependent velocity

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