VIBRATIONS OF AN AXIALLY MOVING BEAM WITH TIME-DEPENDENT VELOCITY

Journal of Sound and <ibration (1999) 227(2), 239}257Article No. jsvi.1999.2247, available online at http://www.idealibrary.com on

VIBRATIONS OF AN AXIALLY MOVING BEAM WITHTIME-DEPENDENT VELOCITY

H. R. OG Z AND M. PAKDEMI0 RLI0

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

(Received 16 September 1998, and in ,nal form 15 March 1999)

The dynamic response of an axially accelerating, elastic, tensioned beam isinvestigated. The time-dependent velocity is assumed to vary harmonically abouta constant mean velocity. These systems experience a coriolis accelerationcomponent which renders such systems gyroscopic. The equation of motion issolved by using perturbation analysis. Principal parametric resonances andcombination resonances are investigated in detail. Stability boundaries aredetermined analytically. It is found that instabilities occur when the frequency ofvelocity #uctuations is close to two times the natural frequency of the constantvelocity system or when the frequency is close to the sum of any two naturalfrequencies. When the velocity variation frequency is close to zero or to thedi!erence of two natural frequencies, however, no instabilities are detected up tothe "rst order of perturbation. Numerical results are presented for di!erent #exuralsti!ness values and for the "rst two modes.

( 1999 Academic Press

1. INTRODUCTION

Due to their technological importance, the vibrations of axially moving materialshave been investigated by many researchers. Threadlines, high-speed magnetic andpaper tapes, strings, power transmission chains and belts, band-saws, "bres, beams,aerial cable tramways and pipes conveying #uid are some of the examples. Ulsoyet al. [1] and Wickert and Mote [2] reviewed the literature on axially movingmaterials. Wickert and Mote [3] investigated the transverse vibrations of travellingstrings and beams. They used travelling string eigenfunctions and introduceda convenient orthogonal basis suitable for discretization. The same authors [4] alsopresented a modal analysis using complex state eigenfunctions and theirconjugates. Ulsoy [5] treated a model for the transverse vibration of an axiallymoving beam which includes elastic coupling between two adjacent spans. Thesystem was then analyzed by using a classical approximate solution method. It wasconcluded that the presence of a beating phenomenon at low transport velocitieswas destroyed by higher velocities and/or tension di!erences in the two spans.Al-Jawi et al. [6}8] "rst investigated the vibration localization phenomenon indual-span axially moving beams. Wu and Mote [9] studied simple torsionparametric resonances and combination torsion-bending parametric resonances in

0022-460X/99/420239#19 $30.00/0 ( 1999 Academic Press

240 H. R. OG Z AND M. PAKDEMI0 RLI0

an axially moving band by an in-plane periodic edge loading that was normal to thelongitudinal axis of the band. A theoretical basis for the analysis of band vibrationand stability was studied by Ulsoy and Mote [10]. The band natural frequencieswere found to decrease with increasing axial velocity at a rate dependent on thewheel support system constant, and to increase with increasing axial tension or&&strain''.

Pakdemirli and Ulsoy [11] investigated principal parametric resonances andcombination resonances for any two modes for an axially accelerating string. Theyfound that for velocity #uctuation frequencies near twice any natural frequency, aninstability region occurs whereas for the frequencies close to zero, no instabilitieswere detected. For combination resonances, instabilities occurred only for those ofadditive type. No instabilities were detected for di!erence-type combinationresonances in agreement with reference [12]. Oz et al. [13] investigated thetransition behaviour from strip to beam for axially moving continua. Anapproximate analytical expression for the natural frequency was given for theproblem. For variable velocity pro"les, stability borders were determinedanalytically. The beam e!ects were studied. Wickert [14] analyzed free non-linearvibrations of a moving beam over the sub- and superharmonic transport speedranges. Pellicano and Zirilli [15] presented a boundary layer solution for theaxially moving beam problem with small #exural sti!nesses. Asokanthan andAriaratnam [16] investigated #exural instabilities in moving bands underharmonic tension. They discussed the e!ects due to damping, mean band speed,and the band compliance on the band stability.

In most of the references given, the transport velocity was taken as constant. Inreality, the systems are exposed to accelerating and decelerating motions. Miranker[17] took a model for the transverse vibrations of a tape moving between a pair ofpulleys and by using a variational procedure derived the equations of motion fortime-dependent axial velocity. Mote [18] investigated the problem of an axiallyaccelerating string with harmonic excitation at one end and determined stability byLaplace transform techniques. Pakdemirli et al. [19] re-derived the equations ofmotion for an axially accelerating string using Hamilton's principle andnumerically investigated the stability of the response using Floquet theory.A sinusoidal variation of the transport velocity, about a mean velocity of zero,was considered in the analysis. Pakdemirli and Batan [20] considered a di!erenttype of velocity variation, namely the periodic, constant acceleration}decelerationtype.

In this study, an Euler}Bernoulli beam having di!erent #exural sti!ness valuesand moving with harmonically varying velocities is considered. The beam is simplysupported at both ends. The equation of motion is derived by following a methodsimilar to that given in reference [14]. A harmonically varying velocity function ischosen. The equation of motion is solved by directly applying the method ofmultiple scales to the partial di!erential system (direct-perturbation method). Forhigher order expansions, solutions obtained by using this direct-perturbationmethod better represent the real behaviour of the system than the common methodof discretization (discretization}perturbation method) [21}28]. In the present case,which contains only two terms in the expansion, the advantage of using the

AXIALLY MOVING BEAM 241

direct-perturbation method is that the equations need not be cast into a formconvenient for orthogonalizing the modes [11]. The natural frequency variationwith velocity for various #exural sti!ness values are determined for the "rst twomodes. Principal parametric resonances, sum and di!erence-type combinationresonances are investigated. For velocity #uctuation frequencies near twice anynatural frequency, an instability region occurs whereas for frequencies close to zero,no instabilities are detected. For combination resonances, instabilities occurredonly for those of additive type. No instabilities are detected for di!erence-typecombination resonances up to the "rst approximation order.

2. APPROXIMATE ANALYSIS

For the axially moving beam illustrated in Figure 1, by following a derivationsimilar to that given in reference [14], it can be shown that the linear,time-dependent, dimensionless equation of motion is

wK#2wR @v#w@v5 #l2fwil#(v2!1)wA"0, (1)

where w is the transverse displacement, v is the axial velocity and lf

is thedimensionless #exural sti!ness. The material particle of the travelling beamexperiences local wK , Coriolis 2wR @v, and centripetal v2wA acceleration components.The boundary conditions are

w(0, t)"w(1, t)"0, wA(0, t)"wA(1, t)"0. (2)

When l2f"0, equation (1) reduces to that of a travelling string. The dot denotes

di!erentiation with respect to time and the prime denotes di!erentiation withrespect to the spatial variable x.

Assuming that the velocity is harmonically varying about a constant meanvelocity v

0, one writes

v"v0#ev

1sinXt, (3)

where e is a small parameter and ev1, which is also small, represents the amplitude

of #uctuations, X is the #uctuation frequency. Unlike in reference [13], l2fis now an

order one term. Substituting equation (3) into equation (1) and keeping terms up tothe "rst order of approximation, one has

wK#2v0wR @#(v2

0!1)wA#l2

fwil#e(v

1X cosXtw@#2v

1sinXtwR @#2v

0v1sinXt wA)"0.

(4)

Figure 1. Axially moving beam with time-dependent velocity.


The direct perturbation method will be applied to equation (4) in search ofsolutions. This method does not require conversion of the equation into otherforms as done in the discretization}perturbation method [11]. Using the method ofmultiple scales [29, 30] and assuming a "rst order expansion, one writes

w(x, t; e)"w0(x,¹0 , ¹1)#ew1 (x,¹0, ¹1)#2 , (5)

where w0 and w1 are the displacement functions at orders 1 and e, ¹0"t and¹1"et are the usual fast and slow time scales. In terms of the new variables, thetime derivatives can be written as

d/dt"D0#eD1#2 , d2/dt2"D20#2eD0D1#2 , (6)

where Dn"L/L¹

n. Substituting equations (5) and (6) into equation (4), separating

terms at each order of e, one obtains

O(1): D20w0#2v0D0w@0#(v20!1)wA0#l2fwil0"0, (7)

O(e): D20w1#2v0D0w@1#(v20!1)wA1#l2fwil1"!2D0D1w0!2v0D1w@0

!2v1 sinX¹0D0 w@0!2v0v1 sin X¹0wA0!Xv1 cosX¹0w@0 (8)

The solution at order 1 can be written as follows:

w0(x,¹0,¹1 ; e)"An(¹1) eiun

¹0>

n(x)#AM

n(¹1) e~*un

¹0>1

n(x), (9)

The spatial functions >n(x) satisfy the equation

l2f>il

n#(v20!1)>A

n#2iv0un

>@n!u2

n>n"0, (10)

with the boundary conditions

>n(0)"0 >

n(1)"0; >A

n(0)"0, >A

n(1)"0. (11)

The solution is

>n(x)"c1n(eib1n

x#C2neib2n

x#C3neib3n

x#C4neib4n

x). (12)

The bin

satisfy the dispersive relation

l2fb4in#(1!v20)b2

in!2u

nv0bin

!u2n"0, i"1, 2, 3, 42 , n"1, 22 . (13)

Applying the boundary conditions to the solution, one obtains the matrix equation

1 1 1 1b21n

b22n

b23n

b24n

eib1n eib2n eib3n eib4n

b21n

eib1n b22n

eib2n b23n

eib3n b24n

eib4n

igjgk

1

C2n

C3n

C

egfgh

c1n"

igjgk

0

0

0

0

egfgh

. (14)

4n


For non-trivial solutions, the determinant of the coe$cient matrix must be zero,which yields the support condition

[ei (b1n#b

2n)#ei (b3n

#b4n

)] (b21n!b2

2n) (b2

3n!b2

4n)

#[ei (b1n#b

3n)#ei (b2n

#b4n

)] (b22n!b2

4n) (b2

3n!b2

1n)

#[ei (b2n#b

3n)#ei (b1n

#b4n

)] (b21n!b2

4n) (b2

2n!b2

3n)"0. (15)

Numerical values of unas well as b

incan be calculated by using equations (13) and

(15). These results are presented in section 5. Using the set of equations in equation(14), one can "nd the coe$cients C

2n, C

3nand C

4nby elimination:

C2n"!

(b24n!b2

1n) (eib3n!eib1n)

(b24n!b2

2n) (eib3n!eib2n)

, C3n"!

(b24n!b2

1n) (eib2n!eib1n)

(b24n!b2

3n) (eib2n!eib3n)

, (16, 17)

C4n"!1!C

2n!C

3n. (18)

Hence, >n(x) now reads

>n(x)"c

1nGeib1nx!

(b24n!b2

1n) (eib3n!eib1n)

(b24n!b2

2n) (eib3n!eib2n)

eib2nx!

(b24n!b2

1n) (eib2n!eib1n)

(b24n!b2

3n)(eib2n!eib3n)

eib3nx

#A!1#(b2

4n!b2

1n) (eib3n!eib1n)

(b24n!b2

2n) (eib3n!eib2n)

#

(b24n!b2

1n) (eib2n!eib1n)

(b24n!b2

3n)(eib2n!eib3n)B eib4n

xH . (19)

At the second order of approximation, one substitutes equation (9) into equation(8). The result is

D20w

1#2v

0D

0w@1#(v2

0!1)wA

1#l2

fwil

1"

!2D1A

n(iu

n>n#v

0>@n)eiun

¹0#2D

1AM

n(iu

n>Mn!v

0>M @n) e~iu

n¹

0

!unv1[A

n> @n(ei(un

#X )¹0!ei(un

!X)¹0 )!AM

n>M @n(ei(X!u

n)¹

0!e~i(X#un)¹

0)]

#iv0v1[A

n>An

(ei(un#X )¹

0!ei(un!X)¹

0 )#AMn>M An

(ei(X!un)¹

0!e~i(X#un)¹

0)]

!

X2

v1[A

n> @n(ei(un

#X)¹0#ei(un

!X)¹0)#AM

n>M @n(ei(X!u

n)¹

0#e~i(X#un)¹

0)]. (20)

Di!erent cases arise depending on the numerical value of velocity variationfrequency. These cases will be treated in the following sections.


3. PRINCIPLE PARAMETRIC RESONANCES

In this section, one assumes that one dominant mode of vibration exists.Depending on the numerical value of frequency, three cases will be investigatedseparately.

3.1. X AWAY FROM 2un

AND 0

In this case, equation (20) becomes

D20w

1#2v

0D

0w@1#(v2

0!1)wA

1#l2

fwil

1

"!2D1A

n(iu

n>n#v

0> @n)eiu

n¹

0#cc#NS¹, (21)

where cc and NS¹ denote complex conjugates and non-secular terms respectively.One can take the solution of equation (21) as

w1(x, ¹

0,¹

1)"/

n(x,¹

1)eiu

n¹

0#= (x,¹0,¹

1)#cc, (22)

The "rst term is related to secular terms and the second term is related tonon-secular terms. If equation (22) is substituted into equation (21), the /

nare found

to satisfy the equation

l2f/iln#(v2

0!1)/A

n#2iv

0u

n/@

n!u2

n/

n"!2D

1A

n(iu

n>n#v

0> @n) (23)

and the boundary conditions are

/n(0)"0, /A

n(0)"0, /

n(1)"0, /A

n(1)"0. (24)

The solvability condition requires (see reference [29] for details of calculatingsolvability conditions)

D1A

n"0. (25)

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

An"A

0. (26)

Hence, solutions are bounded for this case up to O(e).

3.2. X CLOSE TO 0

For this case, the nearness of X to zero is expressed as

X"ep. (27)A similar calculation yields the solvability condition as

D1A

n#(k

1cosp¹

1#k

2sin p¹

1)A

n"0, (28)


where

k1"

Xv1P

1

0

>@n>Mndx

2AiunP1

0

>n>Mndx#v

0P1

0

>@n>MndxB

, k2"

v1AiunP

1

0

>@n>Mndx#v

0P1

0

>An>MndxB

AiunP1

0

>n>Mndx#v

0P1

0

>@n>MndxB

.

(29)

After solving equation (28), one obtains

An"A

0e~(k

1sinp¹

1/p)#k

2cosp¹

1/p). (30)

Since Dsin p¹1D)1 and Dcosp¹

1D)1, the complex amplitudes are bounded in time.

Therefore, there are no instabilities up to this order of approximation.

3.3. X CLOSE TO 2un

In this case, to represent the nearness of velocity variation frequency to two timesone of the natural frequencies, one writes

X"2un#ep, (31)

where p is a detuning parameter. The solvability condition requires

D1A

n#k

0AM

neip¹

1"0, (32)

where k0

is

k0"

G12 (X!2un)P

1

0

>M @n>Mndx!iv

0P1

0

>M An>MndxH

2Giun P1

0

>Mn>ndx#v

0 P1

0

>Mn>@ndxH

v1,

(33)

To perform a stability analysis, one introduces the transformation

An"B

neip¹

1/2. (34)

Bn

can be written in real and imaginary parts in the form

Bn"(bR

n#ibI

n)ej¹

1. (35)

Substituting equation (34) into equation (32) and equation (35) into the resultingequation, and separating real and imaginary parts, one obtains the matrix equation

(j#k0R

) Ak0I!p2B

Ak0I#p2B (j!k

0R)GbRn

bInH"G

00H, (36)


where k0R

and k0I

are the real and imaginary parts of k0

de"ned in equation (33).For a non-trivial solution (bR

nO0, bI

nO0), the determinant of the coe$cient matrix

must be zero:

j2"!(p2/4)#k20R

#k20I

. (37)

Two roots are obtained from equation (37):

j1,2

"GJ!(p2/4)#k20R

#k20I

. (38)

For

!Jk20R

#k20I(p/2(Jk2

0R#k2

0I(39)

the response is unstable whereas it is stable outside this region. Hence the stabilityboundaries are determined by

p1,2

"G2Jk20R

#k20I

. (40)

Inserting p further into equation (31) gives the "nal result for the frequencies:

X"2unG2eJk2

0R#k2

0I. (41)

The two values of X denote the stability boundaries for small e. For a beam withconstant velocity, v

1"0 and hence k

0"0 from equation (33) and no instabilities

arise for this case. When the amplitude of #uctuations l1

increases, the stabilityregions widen.

4. COMBINATION RESONANCES

In this section, it is assumed that there are two dominant modes. Two cases aresigni"cant. The velocity variation frequency may either be nearly equal to the sumof any two modes or to the di!erence of any two modes.

4.1. COMBINATION RESONANCES OF SUM TYPE

Upon taking two dominant modes (i.e. the nth and mth modes)

X"um#u

n#ep, (42)

the solution can be written at O(1) as

w0(x,¹

0,¹

1)"A

n(¹

1)eiu

n¹

0>n(x)#A

m(¹

1) eiu

m¹

0>m(x)#cc, (43)


where>nis de"ned in equation (19). Substituting equation (43) into equation (8) and

using relation (42), one obtains

D20w

1#2v

0D

0w@1#(v2

0!1)wA

1#l2

fwil

1

"M!2D1A

n(iu

n>n#v

0> @n)

#((um!1

2X)>M @

m#(iv

0>M Am)AM

mv1eip¹

1Neiun¹

0

#M!2D1A

m(iu

m>m#v

0>@m)

#((un!1

2X)>M @

n#iv

0>M An

)AMnv1eip¹

1Neium¹

0#cc#NS¹. (44)

Taking the solution for O(e) as

w1(x,¹

0,¹

1)"/

n(x,¹

1)eiu

n¹

0#/m(x,¹

1)eiu

m¹

0#= (x,¹0,¹

1)#cc, (45)

where the "rst two terms are related to secular terms and the third term is fornon-secular terms. Substituting equation (45) into equation (44) and separating thesolutions for nth and mth modes, one obtains

l2f/il

n#(v2

0!1)/A

n#2iv

0u

n/@

n!u2

n/n

"!2D1A

n(iu

n>n#v

0>@

n)#M(u

m!1

2X )>M @

m#iv

0>M AmNAM

mv1eip¹

1 , (46)

l2f/il

m#(v2

0!1)/A

m#2iv

0u

m/@m!u2

m/

m

"!2D1A

m(iu

m>m#v

0>@

m)#M(u

n!1

2X )>M @

n#iv

0>M AnNAM

nv1eip¹

1 . (47)

These two equations can be solved in a way similar to that for the principalparametric resonance case and the complex amplitude modulation equations areobtained as

D1A

n#k

3AM

meip¹

1"0, D1A

m#k

4AM

neip¹

1"0, (48, 49)

where

k3"!

G(um!1

2X)P

1

0

>M @m>Mndx#iv

0P1

0

>M Am>MndxH v

1

2AiunP1

0

>n>Mndx#v

0 P1

0

>@n>MndxB

, (50)

k4"!

G(un!1

2X)P

1

0

>M @n>Mmdx#iv

0P1

0

>M An>Mm

dxH v1

2AiumP1

0

>m>Mm

dx#v0 P

1

0

>@m>Mm

dxB. (51)


For equations (48) and (49), one introduces the transformations

An"B

neip¹

1/2, A

m"B

meip¹

1/2 (52)

and obtains

D1B

n#i

p2Bn#k

3B1

m"0, D

1B

m#i

p2Bm#k

4BMn"0. (53, 54)

The stability may be directly determined from the above equations. One assumesthat equations (53) and (54) possess solutions of the form

Bn"b

nej¹

1, Bm"b

mej1 ¹

1 , (55)

where bn

and bm

are real. Substituting (55) into the equations (53) and (54),taking the complex conjugate of the second equation, one obtains for non-trivialsolutions

j"GJ!(p2/4)#k3k14. (56)

Using de"nitions (50) and (51), one may show that k3k14

is always real. If k3k14(0,

then the system is stable everywhere and if k3k14*0 then the stability boundaries

are determined by the lines

p"G2Jk3k14, (57)

or

X"um#u

nG2eJk

3k14. (58)

Numerical treatment of the above equation is given in the next section.

4.2. COMBINATION RESONANCES OF DIFFERENCE TYPE

Upon assuming m'n without loss of generality, the nearness of frequency to thedi!erence of mth and nth modes is expressed as

X"um!u

n#ep. (59)

A calculation similar to that for the sum-type resonances yields the complexamplitude equations

D1A

n#k

5A

me~ip¹

1"0, D1A

m#k

6A

neip¹

1"0, (60, 61)


where

k5"!

G(um!1

2X)P

1

0

> @m>Mndx!iv

0 P1

0

>Am>MndxHv

1

2A iun P

1

0

>n>Mndx#v

0 P1

0

>@n>MndxB

, (62)

k6"!

G!(un#1

2X)P

1

0

> @n>Mm

dx#iv0 P

1

0

>An>Mm

dxHv1

2Aium P

1

0

>m>Mm

dx#v0 P

1

0

>@m>Mm

dxB, (63)

A suitable transformation for this case is

An"B

ne~ip¹

1/2, A

m"B

meip¹

1/2 . (64)

Substituting into the equations yields

D1Bn!i

p2

Bn#k

5B

m"0, D

1B

m#i

p2

Bm#k

6B

n"0. (65, 66)

One assumes solutions of the form

Bn"b

nej¹

1, Bm"b

mej¹

1 , (67)

For non-trivial solutions,

j"GJ!(p2/4)#k5k6. (68)

From the de"nitions given in equations (62) and (63) it can be shown that k5k6

isalways a negative real number,

k5k6(0. (69)

In equation (68), the region inside of the root is always negative. j is then a pureimaginary number denoting that the solutions are always bounded. In conclusion,for di!erence-type combination resonances, no instabilities arise up to O(e).

5. NUMERICAL EXAMPLES

In this section, numerical plots for the natural frequencies and stability borderswill be presented.

Natural frequencies are found by solving equations (13) and (15) simultaneouslyand plotted in Figures 2 and 3 for the "rst and second modes respectively for sixdi!erent #exural sti!ness values. As seen, with increasing mean velocity, naturalfrequencies decrease. Increasing #exural sti!ness (introducing beam e!ects) on theother hand causes an increase in natural frequencies.

Figure 2. Comparisons of "rst natural frequency values for di!erent #exural sti!nesses.

Figure 3. Comparisons of second natural frequency values for di!erent #exural sti!nesses.


Figure 4. Stable and unstable regions for principle parametric resonances for the "rst mode (n"1,lf"0)2).




Figure 7. Stable and unstable regions for principle parametric resonances for the second mode(n"2, l

f"0)2).



f"0)6).


f"1)0).


Figure 10. Stable and unstable regions for combination resonances of sum type for the "rst twomodes (n"1, m"2, l

f"0)2).


f"0)6).



f"1)0).


In Figures 4}6, stable and unstable regions are plotted for principle parametricresonances case for the "rst mode for three di!erent #exural sti!ness values(l

f"0)2, 0)6 and 1). In all "gures (Figures 4}12), the regions in between the planar

surfaces are unstable whereas the remaining regions are stable. With increasing#exural sti!ness value, the stability regions shift to higher X values. Increasing thevelocity variation amplitude enlarges the stability regions. In Figures 7}9, stableand unstable regions are given for the second mode. Similar conclusions can bedrawn, as written for Figures 4}6.

In Figures 10}12, stable and unstable regions are plotted for combinationresonances of sum-type case for the "rst two modes for three #exural sti!nessvalues. As the #exural sti!ness value increases, the stability regions shift to higherX values. Increasing the velocity variation amplitude causes the stability regions tobe wider.

6. CONCLUDING REMARKS

In this study, the vibrations of an axially moving Euler}Bernoulli beam has beeninvestigated. The velocity is assumed to be harmonically changing about a meanvalue. The method of multiple scales is applied to the equation of motion.Velocity-dependent natural frequencies are found by using a standard root-"ndingalgorithm for di!erent #exural sti!nesses for the "rst two modes.


The in#uence of small #uctuations of velocity on the stability of the system isinvestigated. The boundaries separating stable and unstable regions are calculated.Principal parametric resonances and combination resonances of sum and di!erencetype for any two modes are considered in the analysis. It is found that for velocity#uctuation frequencies near twice any natural frequency, an instability regionoccurs whereas for the frequencies close to zero, no instabilities are detected up tothe "rst order of approximation. For sum-type combination resonances,instabilities do occur. On the contrary, no instabilities are detected fordi!erence-type resonances up to the "rst order of approximation. Boundariesseparating stable and unstable regions are plotted for principle parametric andcombination resonances of sum type. It is shown that beam e!ects cause thestability boundaries to shift to higher frequency values and increasing the velocityvariation amplitude causes the stability regions to be wider.

ACKNOWLEDGMENT

This work is supported by the Scienti"c and Technical Research Council ofTurkey (TUBITAK) under project number MISAG-119.

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VIBRATIONS OF AN AXIALLY MOVING BEAM WITH TIME-DEPENDENT VELOCITY

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