Supercriticalvibrationofnonlinearcoupledmovingbeamsbasedon discrete Fouriertransform

8/9/2019 Supercriticalvibrationofnonlinearcoupledmovingbeamsbasedon discrete Fouriertransform

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Supercritical vibration of nonlinear coupled moving beams based on

discrete Fourier transform

H. Ding a,n, G.C. Zhang a, L.Q. Chen a,b

a Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, Chinab Department of Mechanics, Shanghai University, Shanghai 200444, China

a r t i c l e i n f o

Keywords:

Supercritical

Axially moving beams

Nonlinearity

Coupled vibration

Natural frequency

Discrete Fourier transform

a b s t r a c t

Natural frequencies of nonlinear coupled planar vibration are investigated for axially moving beams in

the supercritical transport speed ranges. The straight equilibrium configuration bifurcates in multipleequilibrium positions in the supercritical regime. The finite difference scheme is developed to calculate

the non-trivial static equilibrium. The equations are cast in the standard form of continuous gyroscopic

systems via introducing a coordinate transform for non-trivial equilibrium configuration. Under fixed

boundary conditions, time series are calculated via the finite difference method. Based on the time

series, the natural frequencies of nonlinear planar vibration, which are determined via discrete Fourier

transform (DFT), are compared with the results of the Galerkin method for the corresponding governing

equations without nonlinear parts. The effects of material parameters and vibration amplitude on the

natural frequencies are investigated through parametric studies. The model of coupled planar vibration

can reduce to two nonlinear models of transverse vibration. For the transverse integro-partial-

differential equation, the equilibrium solutions are performed analytically under the fixed boundary

conditions. Numerical examples indicate that the integro-partial-differential equation yields natural

frequencies closer to those of the coupled planar equation.

Crown Copyright & 2011 Published by Elsevier Ltd. All rights reserved.

1. Introduction

The axially moving beams have received a great deal of

attention lately. This class of systems has several applications,

including robot arms, conveyor belts, high-speed magnetic tapes,

and automobile engine belt. Understanding the vibrations of

axially moving beams are important for the design of the devices.

Recent developments in research on axially moving structures

have been reviewed in Refs. [13].

The analyses for vibration of finite axially moving beam models

have been conducted via various solution techniques, such as the

Galerkin method for the first three frequencies and modes[4], and

the first two natural frequencies [5] under simple supportedboundary conditions; the complex mode method for natural fre-

quencies and modes in the cases of pinnedpinned ends[1,6] and

clampedclamped ends [7,8], and hybrid supports ends [9]; the

perturbation technique for determining approximate natural fre-

quencies in the subcritical speed ranges [10]; the artificial neural

networks technique for the first two natural frequencies [11]; the

differential quadrature method for determining natural frequencies

and the mode shapes for cracked beam [12]. In all of the above

literatures, the natural frequencies of axially moving beams were

calculated from linear governing equation of transverse vibration.

The known exception is that Wickert [13] used a perturbation

method to investigate the fundamental frequencies of axially mov-

ing beams from a nonlinear integro-partial-differential equation of

one-dimensional vibration. It has been observed that the linear

model of the system is not adequate, and effects of nonlinearity are

predominant at critical speeds [13]. However, so far there are very

limited researches on the natural frequencies of nonlinear vibration.

The Fourier transform has been widely used in circuit analysis

and synthesis, from filter design to signal processing and image

reconstruction. The DFT is an approximation of the Fourier trans-form in a digital environment for computing the Fourier transform

numerically on a computer. The natural frequencies of nonlinear

vibration can be computed very efficiently by combining the

numerical solutions (i.e. the discrete Fourier series (DFS)) and the

DFT algorithm. Ding and Chen[14]used the fast Fourier transform

algorithm to study the first two natural frequencies of nonlinear

planar vibration of axially moving beams for subcritical speed. The

present investigation studies the natural frequencies of an axially

moving beam in the supercritical regime, and the effects of the

nonlinearity and the vibration amplitude are determined.

If the geometrical nonlinearity has to be considered, transverse

and longitudinal motions of an axially moving beam are usually

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International Journal of Non-Linear Mechanics

0020-7462/$ - see front matter Crown Copyright & 2011 Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijnonlinmec.2011.09.010

n Corresponding author.

E-mail addresses: [email protected] (H. Ding),

[email protected] (G.C. Zhang),[email protected] (L.Q. Chen).

Please cite this article as: H. Ding, et al., Supercritical vibration of nonlinear coupled moving beams based on discrete Fouriertransform, Int. J. Non-Linear Mech. (2011), doi:10.1016/j.ijnonlinmec.2011.09.010

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coupled. Thurman and Mote[15]first developed the full govern-

ing equations of planar motion and calculated the nonlinear

fundamental frequencies from simplified governing equations.

The nonlinear coupled equation has been used by Wang and Mote

[16], Riedel and Tan[17], Sze et al. [18], and Ding and Chen[5].

However, there are very limited works on coupled vibration of

axially moving beams without any simplified. Chen and Ding used

direct numerical approaches to investigate free vibration[19], forced

vibration[3], and the non-trivial equilibrium[20]of coupled vibra-tion of axially moving beams. In the present paper, the supercritical

natural frequencies in planar motion of an axially moving beam are

directly calculated via combining the finite difference method and

the DFT algorithm.

The nonlinear integro-partial-differential equation and the

nonlinear partial-differential equation for governing the trans-

verse motion of an axially moving beam both can be decoupled

from longitudinal motion under certain conditions. The three

nonlinear vibration models of axially moving beams have been

widely used as summarized in [3,5]. Actually, those models are

also applicable in principle to other gyroscopic continua such as

axially moving strings [21] and belts [22], while some new

technological difficulties have to be overcome. The two transverse

models were compared for free vibration [23]of axially moving

beams. It is found that the predictions made by the two models

are qualitatively the same, but quantitatively different. The two

models were compared with the transverse component of the

coupled equation for free vibration [19], forced vibration [3] in

the subcritical speed ranges, and the non-trivial equilibria [20]

and natural frequencies [5] in the supercritical regime. Both

models yield almost the same precision results for some condi-

tions and the integro-partial-differential equation always gives

better results. The calculation for the supercritical natural fre-

quencies of nonlinear vibration of moving beams is based on the

non-trivial equilibrium. Therefore, the comparison with two

transverse models for natural frequencies in the supercritical

regime should be more complex while the amplitude of vibration

is large.

It is worth noting that the literatures on axially moving

materials in the supercritical ranges are rather limited. Although

increasing the operating speed beyond the critical speed can

increase the efficiency of these systems. Wickert [13] proposed

the non-trivial equilibrium analytical solutions for axially moving

beams with the simple support boundary conditions from the

nonlinear integro-partial-differential equation in the supercritical

regime. In this paper, the non-trivial equilibrium solutions under

fixed boundary conditions are performed analytically from the

same nonlinear equation. Hwang and Perkins [24,25] studied the

effect of an initial curvature due to supporting wheels and pulleys

on the bifurcation and stability of equilibrium in the supercritical

speed regime. Ravindra and Zhu [26] studied a parametrically

excited response as the axial velocity of the beam is varied

beyond a critical value. Pellicano and Vestroni [27]investigatedthe dynamics of a traveling beam subjected to a transverse load

via the Galerkin method when its main parameters vary in the

supercritical velocity range. Ding and Chen [20] used the finite

difference method and the differential quadrature method to

study the non-trivial equilibrium, and used the Galerkin method

[5] to truncate the coupled governing equations without non-

linear parts for computing the natural frequencies of an axially

moving beam under the simple support boundary in the super-

critical regime. But the Galerkin method in Ref. [5]cannot predict

the effect of the vibration amplitude on the natural frequencies.

For the neighboring system comprising an initially straight,

simply supported pipe that conveys fluid, Holmes[28]examined

in detail the symmetric saddle-node bifurcation that occurs at the

critical speed. The present investigation studies the natural

frequencies in planar vibration of an axially moving beam under

the fixed boundary conditions in the supercritical regime, and the

numerical results are compared with those results of the Galerkin

method.

The present paper is organized as follows.Section 2establishes

the coupled governing equation and two equations for transverse

motion of an axially moving beam in the supercritical regime.

Section 3 develops the finite difference schemes to solve the

time series from the coupled equations of planar motion pre-sented in Section 2. Section 4 develops the DFT schemes to

solve the natural frequencies from the numerical results pre-

sented inSection 3, and compares the numerical results with the

Galerkin method. Section 5 compares the coupled equations of

planar motion with two governing equations of transverse motion

via the natural frequencies. Section 6 ends the paper with the

concluding remarks.

2. Mathematical models

The governing equations of nonlinear one-dimensional and planar

free oscillations of an axially moving beam have been derived in[19].

The non-trivial equilibrium solutions and the governing equation of

standard form of continuous gyroscopic systems for integro-partial-differential equation under simple supported boundary conditions in

the supercritical regime have been derived in [13]. Following these

references, one can numerically obtain the non-trivial solutions for

the coupled equation and the two transverse equations.

Consider an axially moving beam of flexural rigidity EI, axial

stiffness EA, and traveling with the uniform constant transport speed

Gbetween two fixed boundaries separated by length L. LetP0be the

initial tension andrAis constant mass per unit length of the beam.The fixed axial coordinate X measures the distance from the left

boundary. The longitudinal displacement and the transverse dis-

placement are denoted by U(X,T) andV(X,T), respectively. Assuming

that the beam has only in-plane motion, the coupled equations for

transverse motion and longitudinal displacement of axially moving

elastic beam can be cast into the dimensionless form[19]:

u,tt 2gu,xtg2u,xxk

21u,xx k

211v,x1 u,xv,xx

u,xxv,x1 u,x2 v,2x

3=2

v,tt2gv,xtg2v,xxk

21v,xxk

2fv,xxxx k

2111 u,x1 u,xv,xx

u,xxv,x1 u,x2 v,2x

3=2 1

where a comma precedingx ortdenotes partial differentiation with

respect tox ort, and the dimensionless variables and parameters as

follows:

v V

L , u

U

L , x

X

L , t t

ffiffiffiffiffiffiffiffiffiffiffiP0

rAL2

s , g G

ffiffiffiffiffiffiffirA

P0

s ,

k1ffiffiffiffiffiffiEA

P0s ,

kfffiffiffiffiffiffiffiffiffiffiEI

P0L2s

2

Equilibrium solutions vx and ux of Eq. (1) satisfy

g2u00k21u00

k211v01u

0v

00u

00v

01 u02 v

023=2

g2v00

k21v00

k2fv4

k2111 u01 u

0v

00u

00v

01 u02 v

023=2

3

where the prime indicates differentiation with respect to x and the

superscript indicates the sense of the equilibrium displacement.

In the present investigation, only the boundary conditions of

the beam fixed at both ends are considered as follows:

u0,t u1,t 0 4

v0,

t v1,

t 0,

v,

x0,

t v,

x1,

t 0 5

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from the DFS of the transverse displacement of the beam center via

the DFT. Fig. 2 illustrates the results of the first two natural

frequencies of the beam, which calculated via the DFT fromFig. 1.

Fig. 3illustrates the effects of the flexural stiffness with k1100

andD0.0001 on the first two natural frequencies.Fig. 3shows that

the fundamental frequencies increase with the growth of the axial

speed for the given flexural stiffness kf. The comparisons also indicate

that the second natural frequencies increase with the growth of the

flexural stiffness kffor the given axial speed. In Ref. [15], Thurman

and Mote also show that the contributions of nonlinear terms in the

equations of motion of an axially moving strip increase as the

transport velocity increases for the fundamental frequencies.

Fig. 4 indicates the effects of the vibration amplitude with

k1100 and kf0.8 on the first two natural frequencies. Fig. 4

shows that the vibration amplitude Dhas little effects on the naturalfrequencies when the vibration is rather small. The comparisons also

indicate that there are significant effects on the first two natural

frequencies for rather large vibration, especially in near-critical-

velocity regime, just like Wickerts inclusion of nonlinearity in

vibration studies is the most important at near-critical speeds[13].

Figs. 5 and 6illustrate the effects of the nonlinear coefficient

with kf0.8 on the first two natural frequencies. In Fig. 5, the

solid lines, the up triangles, and the down triangles, respectively,

stand for the natural frequencies to k175, 100, and 150 with

D0.0001.Fig. 5shows that there are no significant effects on the

first two natural frequencies for the different nonlinear coefficient

k1, especially for the fundamental frequencies. In Fig. 6, g6.0.Fig. 6 shows that the first two natural frequencies decrease with

the growth of amplitude rapidly for rather large vibration. The

comparisons also indicate that the effect of the nonlinear coeffi-

cient on the first two natural frequencies increases with the

growth of vibration amplitude. In Ref. [13], Wickert concluded

0

0.000

0.002

0.004

F()

30 60

Fig. 2. Power spectrum ofFig. 1.

0

5

10

15

20

kf = 0.8

kf = 0.6

kf = 1

Axial speed

Naturalfrequency

3

20

30

40

50

Naturalfrequency

Axial speed

kf = 0.6

kf = 0.8

kf = 1

4 5 6 7 8 9 103 4 5 6 7 8 9 10

Fig. 3. Effects of the flexural stiffness on natural frequencies versus axial speed. (a) The first natural frequencies. (b) The second natural frequencies.

5

0

10

20

Axial speed

Naturalfrequency

35

40

45

D = 0.0001

D = 0.001D = 0.003

D = 0.0001

D = 0.001D = 0.003

Axial speed

Naturalfrequency

6 7 8 9 10 5 6 7 8 9 10

Fig. 4. Effects of the vibration amplitude on natural frequencies versus axial speed. (a) The first natural frequencies. (b) The second natural frequencies.




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that the fundamental frequencies decrease with the growth of the

vibration amplitude and the nonlinear coefficient. This conclusion

is verified by the present investigation at large vibration.

4.2. Comparison with the Galerkin method

In Ref. [5], Ding and Chen determined the first two natural

frequencies of an axially moving beam via the Galerkin method to

truncate the coupled model Eq. (10) without nonlinear parts

under the simple supported boundary in the supercritical regime.

The simply supported boundary conditions of the beam at both

ends are as follows:

u0,

t u1,

t 0 18

v0,t v1,t 0,v,xx0,t v,xx1,t 0 19

The initial conditions for Eq. (10) are chosen as second

eigenfunction of a stationary beam under the simple supported

boundary conditions, namely

ux,0 0,u,tx,0 0 20

vx,0 Dsin2px, v,tx,0 0 21

Fig. 7 illustrates that the comparison with the first two natural

frequencies from the 8-term Galerkin method and the DFT algo-

rithm under the simple supported boundary conditions. In Fig. 7,

kf0.8, k1100, and D0.0001.Fig. 7 shows that two results are

almost overlapped, especially for the fundamental frequencies.

For the fixed boundary conditions, both the trial and weight

functions are chosen as eigenfunctions of a stationary beam

under the boundary conditions (4) and (5), namely, suppose that

the solution to Eq. (10) takes the form:

vx,t Xn

j 1

qvjtfcoshbjxcosbjx zjsinbjxsinhbjxg

ux,t Xn

j 1

qujtfcoshbjxcosbjx zjsinbjxsinhbjxg

zjcoshbjcosbjsinhbjsinbj

, j 1,2,. . ., 8 22

where qvjt and qujt are sets of generalized displacements ofthe beam, and b14.73, b27.8532, b310.9956, b414.1372,

b517.2788, b620.4204, b723.5619, b826.7035. After sub-

stituting Eq. (22) into Eq. (10) without nonlinear parts, the

Galerkin procedure leads to the first few natural frequencies.

Fig. 8 shows the comparison with the first two natural

frequencies from the 8-term Galerkin method and the DFT

algorithm under the fixed boundary conditions. InFig. 8, kf0.8,

k1100, and D 0.0001. The numerical results illustrate that the

first two natural frequencies from both methods are almost

coincided. However, there are small errors. On the whole, it can

be concluded that the outcomes of the 8-terms Galerkin method

are verified by the DFT algorithm. Figs. 4 and 6show that there

are significant effects on the first two natural frequencies for

rather large amplitude vibration. It can be concluded that the

0

10

20

Natura

lfrequency

Natura

lfrequency

Axial speed

k1=75

k1=100

k1=150

k1=75

k1=100

k1=150

5

35

40

45

Axial speed

6 7 8 9 105 6 7 8 9 1 0

Fig. 5. Effects of the nonlinear coefficient on natural frequencies versus axial speed. (a) The first natural frequencies. (b) The second natural frequencies.

0

10

20

k1=75k1=150k1=100

Vibration amplitude

Naturalfrequency

Naturalfrequency

0.000

35

40

45

k1=150k1=100

k1=75

Vibration amplitude

0.001 0.002 0.003 0.004 0.0050.000 0.001 0.002 0.003 0.004 0.005

Fig. 6. Effects of the nonlinear coefficient on the natural frequencies versus vibration amplitude. (a) The first natural frequencies. (b) The second natural frequencies.

H. Ding et al. / International Journal of Non-Linear Mechanics ] (]]]]) ] ]]]]] 5



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Galerkin method is not suitable for the frequencies of large

amplitude vibration.

5. The natural frequencies of transverse motion

and comparisons

5.1. Numerical results of the integro-partial-differential equation

Under certain conditions, the transverse motion can be decoupled

from the longitudinal motion so that a nonlinear equation is obtained

to govern the transverse motion. One of such conditions is the quasi-

static stretch assumption. Under the assumption, Wickert developed

a nonlinear model for transverse motion of axially moving beams

[13]. The nonlinear equation is an integro-partial-differential equa-

tion, namely

v,tt 2gv,xt g21v,xxk

2fv,xxxx

1

2k21v,xx

Z 10

v,2x dx 23

Equilibrium solutions vx for Eq. (23) satisfy

g211

2k21

Z 10

v02

dx

!v

00k2fv

0000 0 24

It is assumed that the solution to Eq. (24) takes the form:

^vx Cxsinkpx 25

The substitution of Eq. (25) into the fixed boundary conditions

Eq. (5) yields

C0 C1 0 26

The following expression satisfy Eq. (26):

Cx Asinkpx 27

where A is a constant. The substitution of Eq. (27) into Eq. (25)

yields

vx Asin2kpx 28

The substitution of Eq. (28) into equilibrium Eq. (24) yields

A 72

kk1p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig212kkfp

2q

29

After substituting Eq. (29) into Eq. (28), leads to the equili-

brium solution for Eq. (23):

vx 72

kk1p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig212kkfp

2q

sin2

kpx, k 1,2,3,. . . 30

Using Eq. (30), one can obtain the first critical speeds of the

fixed boundary conditions as following:

g1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2kfp2

q 31

The substitution vx,t-v7

x vx,t in Eq. (23) yields

v,

tt 2gv,

xt g2

1v,

xxk2

fv,

xxxx

0

10

20DFT

8-term Galerkin truncation

Axial speed

35

40

45DFT


Axial speed

5 6 7 8 9 10 5 6 7 8 9 10

Naturalfrequency

Naturalfrequency

Fig. 8. Comparison with the Galerkin method and the DFT: the fixed boundary conditions. (a) The first natural frequencies. (b) The second natural frequencies.

0

5

10

15

DFT


Natura

lfrequency

Natura

lfrequency

Axial speed

kf = 0.8

2.5

25

30

35

DFT


Axial speed

kf = 0.8

3.0 3.5 4.0 4.5 5.02.5 3.0 3.5 4.0 4.5 5.0

Fig. 7. Comparison with the Galerkin method and the DFT: the simple supported boundary conditions. (a) The first natural frequencies. (b) The second natural frequencies.




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1

2k21v,xx

Z 10

v,2x dx 2

Z 10

v,xv7 0

dx

Z 10

v7 02

dx

!

1

2k21v,

7

xx

Z 10

v,2x dx 2

Z 10

v,xv7 0

dx

! 32

Substitution of Eq. (9) into Eq. (32), the transverse displace-

ment of the beam could be numerically solved via the finite

difference schemes under the boundary conditions Eq. (5) and theinitial conditions Eq. (14) [19]. After substituting the transverse

part of Eq. (22) into Eq. (32) without nonlinear parts, the Galerkin

procedure also leads to the first few natural frequencies. Fig. 9

shows that the comparison with the first two natural frequencies

from the 8-term Galerkin method and the DFT algorithm under

the fixed boundary conditions. In Fig. 9, kf0.8, k1100, and

D0.0001. The numerical results illustrate that the first two

natural frequencies from both methods are almost overlapped.

5.2. Numerical results of the partial-differential equation

Inserting u0 into Eq. (1) and then omitting higher order

nonlinear terms yield the partial-differential equation for trans-verse motion of axially moving elastic beam[20]:

v,tt2gv,xt g21v,xxk

2fv,xxxx

3

2k21v,

2x v,xx 33

Equilibrium solutions vx for Eq. (33) satisfy

k2fv4

g213

2k21v

02

v00

0 34

It is worth noting that one can use the averaged value of the

disturbed tension to replace the exact value if the spatial variation

of the disturbed tension is rather small. In this case, Eq. (23) can

also be obtained from Eq. (33).

Substitution of Eq. (9) into Eq. (34) leads to a set of algebraicequations with respect to vj that can be solved under the boundary

conditions Eq. (5), and the nonlinear Eq. (34) can be solved using an

iterative procedure [20]. The substitution vx,t-v7

x vx,t in

Eq. (23) yields

v,tt 2gv,xt g21v,xxk

2fv,xxxx

3k21v7

0

v7

00

v,x3

2k21v

702

v,xx3k21v7

0

v,xxv,x3

2k21v

700

v,x2

3

2k21v,xxv,x

2 35

The first few natural frequencies can also be determined from

Eq. (35) via the DFT and the Galerkin method, respectively. Fig. 10

shows that the first two frequencies are, respectively, calculated

from the 8-term Galerkin method and the DFT algorithm under

the fixed boundary conditions. In Fig. 10, kf0.8, k1100, and

D0.0001. The numerical results illustrate that the first two

natural frequencies from both methods are almost overlapped.

0

10

20

DFT


Axial speed

Natura

lfrequency

Natura

lfrequency

35

40

45

DFT


Axial speed

5 6 7 8 9 10 5 6 7 8 9 10

Fig. 9. Natural frequencies versus axial speed for Eq. (32). (a) The first natural frequencies. (b) The second natural frequencies.

0

5

10

15

20

25

DFT


Axial speed

Naturalfrequency

Naturalfrequency

35

40

45

50

55

60

DFT


Axial speed

5 6 7 8 9 10 5 6 7 8 9 10

Fig. 10. Natural frequencies versus axial speed for Eq. (35).




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5.3. Comparisons

The first two natural frequencies of nonlinear vibration of axially

moving beams are numerically calculated based on numerical solu-

tion and the DFT algorithm. Based on the natural frequencies of

Eqs. (10), (32), and (35), the differences between the two transverse

models can be investigated in the supercritical regime. Fig. 11illus-

trates the natural frequencies for the three models versus axial speed

with fixed flexural stiffness kf0.8, nonlinear coefficient k1100,and vibration amplitude D0.0001. Fig. 12 illustrates the natural

frequencies for the three models versus flexural stiffness with fixed

g6.0, k1100, and D 0.0001. The numerical results demonstratethat there are certain differences among the three nonlinear models

and the differences increase with the axial speed and decrease with

the flexural stiffness. The comparisons also indicate the results of

Eq. (32) are closer to those of Eq. (10), especially for the first natural

frequencies.

Fig. 13 illustrates the natural frequencies for the three models

versus vibration amplitude with fixed g6.0,kf0.8, and k1100.Fig. 14 illustrates the natural frequencies for the three models versus

0

5

10

15

20

25

Eq. (10)

Eq. (32)

Eq. (35)

Eq. (10)

Eq. (32)

Eq. (35)

Axial speed

35

40

45

50

55

60

Axial speed

Naturalfrequency

Naturalfrequency

5 6 7 8 9 10 5 6 7 8 9 10

Fig. 11. Natural frequencies calculated from Eqs. (10), (32), and (35) versus axial speed. (a) The first natural frequencies. (b) The second natural frequencies.

0.0

0

5

10

15

Eq. (10)

Eq. (32)

Eq. (35)

Eq. (10)

Eq. (32)

Eq. (35)

Naturalfre

quency

Naturalfre

quency

Flexural stiffness

0

10

20

30

40

50

Flexural stiffness

0.5 1.0 0.0 0.5 1.0

Fig. 12. Natural frequencies calculated from Eqs. (10), (32), and (35) versus flexural stiffness. (a) The first natural frequencies. (b) The second natural frequencies.

0.000

0

5

10

15

Vibration amplitude

35

40

45

Vibration amplitude

Eq. (10)Eq. (32)

Eq. (35)

Eq. (10)Eq. (32)

Eq. (35)

Naturalfrequency

Naturalfrequency

0.001 0.002 0.003 0.000 0.001 0.002 0.003

Fig. 13. Natural frequencies calculated from Eqs. (10), (32), and (35) versus vibration amplitude. (a) The first natural frequencies. (b) The second natural frequencies.




9/10

nonlinear coefficient with fixed g6.0,kf0.8, and, D 0.001. Thenumerical results demonstrate that there are certain differences

among the three nonlinear models and the results of Eq. (32) are

closer to those of Eq. (10), especially for the first natural frequencies.

Almost like the conclusions in the subcritical transport speed

ranges in Ref.[14]and in the supercritical regime in Ref. [5], in the

view of the natural frequencies in the supercritical regime, while

all three models of axially moving beams compared predicted

same qualitative behavior except the second natural frequencies

versus axial speed, as there are differences between them quan-

titatively. The calculation for the supercritical natural frequencies

of nonlinear vibration of moving beams is based on the non-trivial

equilibrium. However, the comparisons between the two non-

linear models of large transverse vibration of axially moving

beams in the supercritical regime are much more complex than

the situations in the subcritical speed range.

6. Conclusions

It will be important to know its natural frequencies for accurate

prediction of the dynamic characteristics and stability for an axially

moving beam. Natural frequencies of nonlinear planar vibration are

studied for axially moving elastic beams in the supercritical regime

via combining numerical solution and the DFT algorithm. The non-

trivial equilibrium equations of three nonlinear models are solved

via the finite difference scheme. For motion about each bifurcated

solution, those equations are cast in the standard form of continuous

gyroscopic systems by introducing a coordinate transform. The time

histories of transverse displacements of the beam center are solved

via the finite difference scheme under the fixed boundary condi-

tions. The DFT transforms the data sequence to the frequency

domain from the time domain. The natural frequencies are com-pared with the results of the Galerkin method for the corresponding

governing equations without nonlinear parts. Two nonlinear equa-

tions for the transverse vibration, namely, an integro-partial-

differential equation and a partial-differential equation are reduced

from the governing equations of coupled planar vibration. The two

models were compared with the transverse component of the

coupled model via the natural frequencies in the supercritical regime.

The investigation leads to the following conclusions:

(1) The fundamental frequencies increase with the axial speed and

the second natural frequencies increase with the flexural stiff-

ness. The vibration amplitude and the nonlinear coefficient have

little effects on the natural frequencies for small vibration while

there are significant effects for large vibration, and the first two

natural frequencies decrease with the vibration amplitude and

the nonlinear coefficient for large vibration.

(2) The outcomes of the 8-terms Galerkin method are verified by

the DFT algorithm for small vibration and the Galerkin

method is not suitable for the frequencies of axially moving

beams for large vibration in the supercritical regime.

(3) There are certain differences among the three models with

the changing axial speed, flexural stiffness, vibration ampli-

tude, and nonlinear coefficient. The differences decrease with

the flexural stiffness and increase with the axial speed.

(4) In the view of the natural frequencies, the nonlinear integro-

partial-differential equation yields the results closer to those

from the coupled equations.

Acknowledgments

This work was supported by the National Natural Science

Foundation of China (No. 10902064), the National OutstandingYoung Scientists Fund of China (No. 10725209), Shanghai Rising-

Star Program (No. 11QA1402300), Shanghai Subject Chief Scientist

Project (No. 09XD1401700), Shanghai Leading Talent Program,

Shanghai Leading Academic Discipline Project (No. S30106), and

the program for Changjiang scholars and Innovative Research Team

in University (No. IRT0844).

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Please cite this article as: H. Ding, et al., Supercritical vibration of nonlinear coupled moving beams based on discrete Fourierf i h (20 ) d i 0 0 6/j ij li 20 09 0 0

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