Natural frequencies of a super-critical transporting Timoshenko beam

Accepted Manuscript Natural frequencies of a super-critical transporting Timoshenko beam Hu Ding, Xia Tan, Earl H. Dowell PII:

S0997-7538(16)30502-2

DOI:

10.1016/j.euromechsol.2017.06.007

Reference:

EJMSOL 3448

To appear in:

European Journal of Mechanics / A Solids

Received Date: 18 December 2016 Revised Date:

4 May 2017

Accepted Date: 12 June 2017

Please cite this article as: Ding, H., Tan, X., Dowell, E.H., Natural frequencies of a super-critical transporting Timoshenko beam, European Journal of Mechanics / A Solids (2017), doi: 10.1016/ j.euromechsol.2017.06.007. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Natural frequencies of a super-critical transporting Timoshenko beam Hu Ding a,b*, a

Earl H. Dowell c

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China

Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China c

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b

Xia Tan a and

Duke University, Durham, NC, 27708, USA

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Abstract: Super-critical transporting continua are often discussed in the context of an Euler beam model. Here Timoshenko beam theory is applied to study free vibration of high-speed transporting

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continua. Therefore, effects of rotary inertia and shear deformation on transverse vibration of super-critical transporting beams are discovered for the first time. The Galerkin method is applied to solve natural frequencies with the simply supported boundary conditions. Meanwhile, the weighted residual method (WRM) is employed to deal with the fixed ends. The natural frequencies of

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super-critical continua are verified by using the discrete Fourier transform (DFT). In the super-critical regime, the straight configuration of the beam loses its stability. The buckling

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configurations, due to the nonlinear stiffness, are deduced from the nonlinear static equilibrium equation. Furthermore, the governing equation for the transverse vibration of the super-critical

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transporting Timoshenko beam is derived based on the buckling shape. Time histories are calculated by using the finite difference method (FDM). Furthermore, natural frequencies of nonlinear free vibration are determined by using the DFT. In addition, the effect of nonlinear stiffness on the natural frequencies is discovered. On the other hand, a partially linearized equation is deduced by regarding buckling configurations and its spatial derivatives as constant coefficients. Then, natural frequencies are extracted by using the Galerkin method. The two different approaches are compared. *

Corresponding author. Tel.: +86 02156337273. Fax: 86-21-56333085;

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

ACCEPTED MANUSCRIPT Numerical results show that the rotary inertia and the shear deformation significantly affect vibration characteristics of the super-critical transporting beam for some configurations. Moreover, comparisons with an Euler-Bernoulli beam model reveal that the fundamental frequency of the

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Timoshenko beam is higher when the speed is slightly greater than the critical value. Keywords: Nonlinear vibration; transporting beam; Timoshenko theory; super-critical; natural frequency

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1. Introduction

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The main goal of this work is to reveal the effects of rotary inertia and the shear force on super-critical vibration characteristics of a transporting beam. As the mechanical model of many engineering components, axially transporting beams have been widely studied in the past half century [Chen and Wang, 2009; Ding and Chen, 2008; Marynowski, 2006 and 2010; Mote and

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Naguleswaran, 1966; Pakdemirli and Öz, 2008; Sandilo and van Horssen, 2012; Yao et al. 2012; Yang et al. 2010 ]. It is well-known that the dynamics characteristics of transporting systems are significantly affected by the axial speed [Chen and Yang, 2006; Ding et al. 2016; Ding and Zu,

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2014; Metrikine and Dieterman, 1999; Wang et al. 2016; Wickert, 1992]. Chen detailly reviewed on

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advances in the nonlinear vibration of transporting continua, which was important to reveal the progress in this subject [Chen, 2005]. The fundamental frequency becomes zero at the critical speed. Then because of the nonlinear stiffness of the transporting beam, the trivial equilibrium position is replaced by buckling configurations for higher speeds. Then the transporting beam vibrates around the buckling configurations. Therefore, the dynamics of the axially transporting beam is dominated by the nonlinear extensional stiffness. Although the vibration of the super-critical transporting beam has shown rich nonlinear characteristics, the nonlinear vibration of high-speed transporting systems offers new opportunities for research.

ACCEPTED MANUSCRIPT The buckling configurations of the super-critical transporting systems have been studied by several authors, e.g. an axially-moving string on an elastic foundation [Parker, 1999], axially moving beams with elastically restrained in horizontal translation by a spring [Hwang and Perkins, 1992], beams with simply supported ends and the fixed-fixed ends [Ding and Chen, 2011], and

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hybrid boundary conditions [Ding et al. 2011]. The free vibration characteristics of the transporting systems at super-critical speed have been investigated by using a perturbation theory [Wickert,

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1992], asymptotic analysis for small curvature[Hwang and Perkins, 1992], the Galerkin method [Ding and Chen, 2010], and the discrete Fourier transform in conjunction with the finite difference

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method [Ding et al. 2012]. Moreover, chaotic responses of parametrically excited transporting beams in the super-critical regime have been discussed by using the 1-term Galerkin method [Ravindra and Zhu, 1998], and compared with results from the differential quadrature method and the Galerkin truncation with various terms [Ding et al. 2014]. Furthermore, for the forced vibration

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of super-critical transporting beams, steady-state periodic responses have been studied by the multiple scales method and the finite difference method [Ding et al. 2012], and the chaotic

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responses have been investigated by using the Galerkin method [Pellicano and Vestroni, 2002; Ghayesh et al. 2012]. One thing should be mentioned. All of the above-mentioned papers on axially

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transporting beams adopted the Euler-Bernoulli beam theory. Therefore, the effects of the shear force and the rotary inertia on the free vibration characteristics of the super-critical transporting systems have not been discovered. In other words, the Timoshenko beam theory has not been applied to study the super-critical free vibration of transporting systems. On the other hand, the Timoshenko model has been used to reveal the dynamics of the transporting beam in the sub-critical speed range. Free vibration of a transporting Timoshenko beam has been studied by using the spectral element models and the conventional finite element method

ACCEPTED MANUSCRIPT [Lee et al. 2004] and the modal analysis [Tang et al. 2009]. The effects of rotary inertia on the eigenvalues of the axially moving beam have been treated [Chang et al. 2010]. Chen et al. (2010) performed pioneering investigations on accelerating viscoelastic Timoshenko beams. Steady-state responses of the sub-critical transporting Timoshenko beam have been surveyed based on the

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Galerkin truncation [Ghayesh and Balar, 2010]. In Ref. [Tang et al. 2009], steady-state responses of the forced vibration of the transporting viscoelastic Timoshenko beam with 3:1 internal resonances

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are also studied by using the multi-scale method. For two classic boundary conditions, forced vibrations also were studied by using the generalized integral transform technique to find a

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semi-analytical numerical solution [An and Su, 2014]. By using symbolic computation, the pressure distribution of the foundation is studied for an axially moving Timoshenko beam [Cojocaru et al. 2010]. These researchers have found that the influence of the shear force and the rotary inertia should not be neglected for studying the nonlinear vibration of the transporting systems. In the

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super-critical regime, bifurcation and chaos of the transporting viscoelastic Timoshenko beam has been numerically investigated for three-dimensional nonlinear planar dynamics [Ghayesh and

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Amabili, 2013] including parametric excitations with [Yan et al. 2015] and without external excitation [Yan et al. 2014]. Ding et al. derived the non-trivial equilibrium of the super-critical

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transporting Timoshenko beam [Ding et al. 2016]. These above-mentioned papers all revealed that the shear deformation and rotary inertia significantly affect the super-critical nonlinear dynamics of the axially transporting continuum. However, published papers on nonlinear vibration of transporting Timoshenko beams in the super-critical speed range are still very limited. In order to more accurately investigate the nonlinear dynamics of the super-critical transporting systems, the Timoshenko beam model is applied in the present work. The buckling configurations are derived. Then, the governing equation of the transverse vibration around the

ACCEPTED MANUSCRIPT buckling shape is established through coordinate transformation. In conjunction with the FDM, natural frequencies of nonlinear free vibration of the super-critical transporting Timoshenko beam are obtained by using the DFT. Moreover, natural frequencies are calculated by using the Galerkin method or the weighted residual method through a partially linearized equation. These two

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approaches show the influences of the shear deformation, rotary inertia, supported ends and beam theories on natural frequencies of the transverse vibration of the super-critical transporting beam.

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2. Mathematical Model and Buckling Configuration

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Fig. 1 shows the mechanics schematic of an axially transporting continuum. V(X,T) stands for the transverse vibration displacement, where X represents the distance from the left end of the beam, and T represents the time coordinates. Moreover, the beam axially transports with the constant speed Γ. Length L is the distance between two ends. Furthermore, initial axial tension of

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transporting continuum is denoted as P0.

Fig. 1. Schematic of a transporting beam

This paper focuses on the nonlinear free vibration characteristics of the transporting beam in the super-critical regime. Moreover, the influence of the rotary inertia and the shear force is considered. Therefore, the Timoshenko beam theory is adopted to derive the dynamics model of the transporting continuum. According to the Hamilton principle, with small but finite stretching and in-plane motion assumptions, the governing nonlinear dimensionless equation of transverse vibration of the transporting beam is [Lee et al. 2004; Ding et al. 2016]

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v,tt +2γ v, xt + (κγ 2 − 1) v, xx + k1 (ϕ , x −v, xx ) =

1

1 2 k N v, xx ∫ v,2x dx， 2 0

k2 (ϕ ,tt +2γϕ , xt +γ 2ϕ , xx ) − kf2ϕ , xx + k1 (ϕ − v, x ) = 0

(1)

where dimensionless variables x, t and v(x,t) are defined as v x t , x ↔ ,t ↔ L L L

P0 ρA

(2)

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v↔

and φ(x,t) represents the deflection curve caused by the bending deformation alone, the comma preceding variable represents partial differentiation, dimensionless parameters are introduced as

ρA P0

, k1 =

k0GA I , k2 = 2 , k N = P0 AL

EA , kf = P0

EI , κ = 1 −η , P0 L2

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γ =Γ

(3)

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η = 1 1 + ks ( 2 EA L )  , k0 = ( 5 + 5µ ) ( 6 + 5µ )

where ρ and A, respectively, are the mass density and the cross-sectional area. E, I, and EI, respectively, denote the modulus of elasticity, the area moment of inertia of the cross-section about the neutral axis and the bending rigidity of the transporting beam. G stands for the shearing

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modulus of the beam. Moreover, dimensionless parameters k1 and k2, respectively, represent the effects of the shear force and the rotary inertia of the transporting beam. kN accounts for the effect of the nonlinear stiffness of the transporting beam. kf represents the bending stiffness. ks is the axial

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support rigidity at the right end.μ is Poisson ratio. k0 is called the shape factor of cross-section,

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also known as the section shear correction factor in Timoshenko's beam theory, used to correct the assumption that the shear stress and the shear strain are evenly distributed across the cross section. The value of the shape factor is determined by the shape of the cross-sectional area and the Poisson's ratio of the material [Hutchinson, 2001]. In this work, the spatial variation of the disturbed tension is assumed to be rather small, then the exact value of the disturbed tensions can be replaced by the averaged value. The assumption approach is also referred as Kirchhoff's approach [Suweken and Van Horssen, 2003]. Then the integro-partial-differential nonlinear term in Eq. (1) can be introduced by omitting the high order nonlinear term.

ACCEPTED MANUSCRIPT The simply supported boundary conditions at both ends are considered as v ( 0, t ) = 0, v (1, t ) = 0,

(k

2 f

− k 2γ 2 ) ϕ , x ( 0, t ) − k 2γϕ ,t ( 0, t ) = 0, ( kf2 − k 2γ 2 ) ϕ , x (1, t ) − k 2γϕ ,t (1, t ) = 0

(4)

It is well-known that the equilibrium buckling configurations are not related to the time

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coordinate. Therefore, buckling configurations vˆ ( x ) and ϕˆ ( x ) satisfy the following ordinary differential equations and boundary conditions with the time derivatives deleted in Eqs. (1) and (4)

(κγ − 1) vˆ′′ + k (ϕˆ ′ − vˆ′′) = 12 k vˆ′′∫ vˆ′ dx, ( k γ − k ) ϕˆ ′′ + k (ϕˆ − vˆ′) = 0, 2 N

1

2

2 f

2

1

1

0

2

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2

vˆ ( 0 ) = vˆ (1) = 0, ϕˆ ′ ( 0 ) = ϕˆ ′ (1) = 0

(5)

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Corresponding to the first critical speed, the non-trivial buckling configurations of the axially transporting Timoshenko beam with the simply supported ends are deduced as [Ding et al. 2016] 2 vˆ ( x ) = ± kN π

k1 (1 − κγ 2 ) − π 2 ( k2γ 2 − kf2 )(1 + k1 − κγ 2 ) π 2 ( k 2γ 2 − kf2 ) − k1

sin ( πx ) ,

(6)

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k1 (1 − κγ 2 ) − π 2 ( k 2γ 2 − kf2 )(1 + k1 − κγ 2 ) 2 ϕˆ ( x ) = ± cos ( πx )  kf2 − k 2γ 2 2  π 2 ( k2γ 2 − kf2 ) − k1 k N 1 + π  k1  

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3. The Galerkin Truncation

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For vibration about each non-trivial buckling configuration, the governing equation of super-critical transporting Timoshenko beam is derived by introducing a coordinate transform. Specifically, substituting v ( x, t )→ vˆ ( x ) + v ( x, t ) into Eq. (1) yields

v,tt +2γ v, xt + (κγ 2 − 1 − k1 ) v, xx + k1ϕ , x = 1 1 1 1  1 1  1 2 k N v, xx  ∫ v,2x dx + ∫ vˆ,2x dx + 2 ∫ v, x vˆ, x dx  + k N2 vˆ, xx  ∫ v,2x dx + 2 ∫ v, x vˆ, x dx ， 2 0 0 0 0  2 0 

(7)

k2ϕ ,tt +2k2γϕ , xt + ( k2γ 2 − kf2 ) ϕ , xx + k1 (ϕ − v, x ) = 0

Due to the results based on v(x, t )→ vˆ − (x ) + v(x, t ) are the exact same, only the results of v(x, t )→ vˆ + (x ) + v(x, t ) are shown in the following study. In order to extract the natural frequencies of

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the super-critical vibration of the transporting Timoshenko beam, vˆ ( x ) and its various spatial derivatives are regarded as constant coefficients with respect to time. Therefore, the governing equation (7) is partially linearized as

k2ϕ ,tt +2k2γϕ , xt + ( k2γ − k ) ϕ , xx + k1 (ϕ − v, x ) = 0 2

2 f

(8)

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1 1  2 1 2 2  2 v,tt +2γ v, xt +  κγ − 1 − k1 − k N ∫ vˆ, x dx  v, xx + k1ϕ , x = k N vˆ, xx ∫ v, x vˆ, x dx， 2 0 0  

Based on Eq. (8), the natural frequencies of the super-critical vibration can be calculated by using

and Balar, 2010; Chen et al. 2007; Kang et al. 2013] n

n

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the Galerkin method. Therefore, the solutions to Eq. (8) are supposed to take the form as [Ghayesh

v ( x, t ) = ∑ q ( t ) sin ( jπx ) , ϕ ( x, t ) = ∑ qϕj ( t ) cos ( jπx ) , n = 1, 2,⋯ j =1

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v j

j =1

(9)

where q vj (t ) and q uj (t ) are generalized coordinates. Substituting Eq. (9) into Eq. (8) leads to 1

n

n

j =1

j =1

n

∑ qɺɺvj ( t ) sin ( jπx ) + 2πγ ∑ qɺ vj ( t ) j cos ( jπx ) − πkN2 vˆ,+xx ∫ ∑ jqvj ( t ) cos ( jπx ) vˆ,+x dx 0 j =1

n n   1 −  κγ 2 − 1 − k1 − k N2 ∫ vˆ, +x 2 dx  π 2 ∑ q vj ( t ) j 2 sin ( jπx ) − πk1 ∑ qϕj ( t ) j sin ( jπx ) = 0, 2 0 j =1   j =1

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1

k2 ∑ qɺɺϕj ( t ) cos ( jπx ) − 2πk2γ ∑ qɺ ϕj ( t ) j sin ( jπx ) − ( k2γ 2 − kf2 ) π 2 ∑ qϕj ( t ) j 2 cos ( jπx ) n

n

j =1

n

j =1

j =1

n

(10)

n

j =1

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+ k1 ∑ qϕj ( t ) cos ( jπx ) − πk1 ∑ q vj ( t ) j cos ( jπx ) = 0 j =1

n

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Then, the Galerkin truncation method yields the following ordinary differential equations 1

1

n

∑ qɺɺ ( t )∫ sin ( jπx ) sin ( iπx )dx + 2πγ ∑ qɺ ( t ) j ∫ cos ( jπx ) sin ( iπx )dx j =1

v j

j =1

0

v j

0

n  2 2 1  n v 1 2 +2 ϕ 2 −  π  κγ − 1 − k1 − k N ∫ vˆ, x dx  ∑ q j ( t ) j + k1π ∑ jq j ( t )  ∫ sin ( jπx ) sin ( iπx )dx 2 0 j =1  j =1    0 1

1 n 1  − πk N2 ∑ jq vj ( t ) ∫ vˆ,+xx  ∫ cos ( jπx ) vˆ,+x dx  sin ( iπx ) dx = 0, j =1 0 0  n n  n ϕ 1  n ϕ  v 2 2 2 ϕ 2 ɺɺ k q t + k q t − π jq t − k γ − k π q t j ( ) ( ) ( ) ( ) ( )  2∑ j  ∫ cos ( jπx ) cos ( iπx )dx  ∑ ∑ 1∑ j j 2 f j j =1 j =1  j =1  0  j =1  n

1

j =1

0

− 2πγ k2 ∑ qɺ ϕj ( t ) j ∫ sin ( jπx ) cos ( iπx )dx = 0

(11)

ACCEPTED MANUSCRIPT Eq. (11) is written as the following matrix-vector form

ɺɺ + Gqɺ + Kq = 0 Mq

(12)

where

 G v  ,G =  Mϕ  

 K vv   ,K =  ϕ Gϕ  K v

1

1 M ijv = ∫ sin ( jπx ) sin ( iπx ) dx = δ ij 2 0 1

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1 M ijϕ = k2 ∫ cos ( jπx ) cos ( iπx ) dx = δ ij 2 0

K ϕv   K ϕϕ 

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 qv  M v q =  ϕ ,M =  q  

4γ ij ( i 2 − j 2 ) , i ≠ j G = 2γ jπ ∫ cos ( jπx ) sin ( iπx )dx =  , i + j = even 0  0 1

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v ij

4 j 2 Gij = −2k2γ jπ ∫ sin ( jπx ) sin ( iπx )dx = − k2γ   0 1

ϕ

(j

2

− i2 )

0

(13)

(14)

(15)

(16)

,i ≠ j , i + j = even

(17)

K vijv = − j 2 π 2 (κγ 2 − 1 − k1 − A ) ∫ sin ( jπ x ) sin ( iπx ) dx 1

0

1   2 − ∫ sin ( iπx )  −4 Ajπ sin(πx) ∫ cos ( jπx ) cos(πx)dx  dx 0 0  

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1

(18)

1

K

v ϕ ij

1 = − k1 jπ ∫ sin ( jπx ) sin ( iπx )dx = − k1 jπδ ij 2 0

(19)

−k1 jπ δ ij 2

(20)

1

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ϕ K vij = − k1 jπ ∫ cos ( jπx ) cos ( iπx )dx = 0

1

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Kϕϕij = − ( k2γ 2 − kf2 ) j 2 π 2 ∫ cos ( jπx ) cos ( iπx )dx 1

0

+ k1 ∫ cos ( jπx ) cos ( iπx )dx =

− ( k2γ 2 − kf2 ) j 2 π 2 +k1

0

2

δ ij

(21)

Moreover, q(t ) is defined as

q(t ) = Qe i ωt

(22)

where Q is constant vector. Substitution of Eq. (22) into Eq. (12) provides a set of coupled algebraic equations  −ω 2 M + i ω G + K  Q = 0

(23)

ACCEPTED MANUSCRIPT The determinant of coefficients of Eq. (23) is required to be zero for existence of non-triviality solutions. Therefore, the natural frequencies ω are calculated from the following equation −ω 2 M + i ωG + K = 0

(24)

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4. Natural Frequencies In the all following study, a belt with rectangular cross section is considered as the engineering prototype of the axially transporting beam. The width is 0.0155 m. The height is 0.018

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m. The density ρ=1200 kg/m3. Furthermore, the length of the transporting beam is L=0.15 m. The initial tension P0 is set as 80 N. The Young's modulus E=200 MPa. The Poisson ratio of the

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transporting beam equals to 0.48. The shape factor of cross-section k0=0.8809 [Hutchinson, 2001]. Therefore, the shearing modulus G is calculated as 68 MPa. The value of those dimensionless parameters defined by Eq. (3) are calculated as kf=0.9149, kN =26.4102, k1=207.5893, k2=0.0012,. Moreover, axial support at the ends is set to be completely inelastic. Based on these dimensionless

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parameter values, dimensionless super-critical transporting speed of the Timoshenko beam with simply supported ends is calculated as γ=3. Therefore, the super-critical speed of the beam equals to

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46.37 m/s. If there is no special designation, the transporting beam is considered with these above-mentioned values in the all following numerical simulations.

9

50 45

2-term Galerkin m ethod 4-term Galerkin method 8-term Galerkin method 7-term Galerkin method

6

3

4

5

γ

2-term Galerkin method 4-term Galerkin method 8-term Galerkin method 7-term Galerkin method

55

ω2

ω1

12

3

60

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15

6

7

(a) The first natural frequency

40 35 30

8

3

4

5

γ

6

7

(b) The second natural frequency

Fig. 2. Natural frequencies which are calculated by using 2, 4, 7 and 8-term Galerkin method

8

ACCEPTED MANUSCRIPT The first two natural frequencies calculated from a 2, 4, 7and 8-term Galerkin method are compared in Fig. 2. As it is seen from Fig. 2, the natural frequencies of the axially transporting Timoshenko beam increase with the translating speed in a certain speed range. Nevertheless, the fundamental frequency slightly decreases with further increase of the axial speed. Furthermore, the

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comparison in Fig. 2 demonstrates that the first two natural frequencies, calculated by using four terms Galerkin method, are almost same as those results when the first seven or the first eight terms are kept in Eq. (9), especially for the fundamental frequency. On the other hand, the fundamental

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frequency based on two terms Galerkin truncation only coincides with those of 7-term and 8-term

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truncations in a very narrow speed range. Therefore, four terms Galerkin truncation provides convergent results for the first two natural frequencies of the super-critical transporting beam. As a consequence, only the first four terms are retained in Eq. (9) in the following computation. Effects of bending stiffness and axial transporting speed on the first two natural frequencies

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are presented in Figs. 3 and 4. As shown in Figs. 3 and 4, the natural frequencies of the super-critical vibration of the transporting Timoshenko beam are very sensitive to the bending stiffness and transporting speed. Furthermore, the fundamental frequency of the super-critical

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transporting beam may become smaller with the higher bending stiffness. 12

38

4-term Galerkin method

4

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10

2

k f=0.6 0

1

kf=1.0

32 30

ω2

ω1

6

0

34

k f=0.8

8

4-term Galerkin method

36

28

k f=0.8

26 24 22

k f=1.0 2

γ

3

4

(a) The first natural frequency

k f=0.6

20

5

18

0

1

2

γ

3

4

(b) The second natural frequency

Fig. 3. Effects of the bending stiffness of the transporting beam versus axial speed

5

ACCEPTED MANUSCRIPT 35

12 4-term Galerkin method

25

ω1

ω2

8 6

20

γ=3.5 γ=4.0 γ=4.5

4

γ=3.5 γ=4.0 γ=4.5

15 10

2 0 0.2

4-term Galerkin method

30

0.4

0.6

kf

0.8

5 0.2

1.0

0.6

kf

0.8

1.0

(b) The second natural frequency

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(a) The first natural frequency

0.4

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10

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Fig. 4. Effects of axial speed of the transporting beam versus the bending stiffness The influences of the rotary inertia and the shear force of the transporting beam on the super-critical characteristics are presented in Figs. 5 and 6, respectively. Figs. 5 and 6 illustrate that the second-order natural frequency of the super-critical transporting beam increases with the

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increasing shear force and the decreasing rotary inertia. With the increasing axial transporting speed, the frequencies are more sensitive to the rotary inertia and the shear force. Furthermore, the second

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frequency is more sensitive to those system parameters than the first frequency.

ω1

9 8

4-term Galerkin method

30

4-term Galerk in method

29

ω2

10

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11

28

γ =3.5 γ =4.0 γ =4.5

7

γ =3.5 γ =4.0 γ =4.5

27 26

6 100

150

200

k1

250

(a) The first natural frequency

300

100

150

200

k1

250

(b) The second natural frequency

Fig. 5. Effects of the shear force and the axial speed of the transporting beam

300

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12

30.0 4-term Galerkin method

11 10

29.0

ω2

9

28.5

γ =3.5 γ =4.0 γ =4.5

8

28.0

7

27.5 6 0.0005

0.0010

0.0015

k2

0.0020

0.0025

0.0030

4-term Galerkin method

0.0005

0.0015

k2

0.0020

0.0025

0.0030

(b) The second natural frequency

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(a) The first natural frequency

0.0010

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ω1

γ=3.5 γ=4.0 γ=4.5

29.5

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Fig. 6. Effects of rotary inertia and the axial speed of the transporting beam

5. Verification by Discrete Fourier Transform

In this section, natural frequencies of the high-speed transporting Timoshenko beam are numerically obtained by using the discrete Fourier transform in conjunction with the finite

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difference method. At first, the governing equation (7) is numerically calculated to obtain time histories by applying the FDM [Ding et al. 2012]. Equispaced mesh grid, L×T, is introduced as

( i = 0,1, 2,...l, h = 1 l ) , t j = jτ ( j = 0,1, 2,....)

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xi = ih

(25)

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where τ is time step and h is space step. Moreover, vˆi denotes as the function values vˆ ( x ) at x=xi. The following difference approximations is adopted

∂vij ∂ϕ j 1 j 1 ≈ vi +1 − vij−1 ) , i ≈ ϕi j+1 − ϕi j−1 ) , ( ( ∂x 2h ∂x 2h 2 j ∂ vi ∂ 2ϕi j 1 j 1 j j ≈ v − 2 v + v , ≈ 2 (ϕi j+1 − 2ϕi j + ϕi j−1 ) , i i −1 ) 2 2 ( i +1 2 ∂x h ∂x h 3 j ∂ vi 1 ≈ 3 ( vij+ 2 − 2vij+1 + 2vij−1 − vij− 2 ) 3 ∂x 2h

∂vij 1 ≈ ( vij +1 − vij −1 ) , ∂t 2τ 2 j ∂ vi ∂ 2ϕi j 1 1 j +1 j j −1 ≈ v − 2 v + v , ≈ 2 (ϕi j +1 − 2ϕi j + ϕi j −1 ) ( ) i i i ∂t 2 τ 2 ∂t 2 τ

(26)

(27)

ACCEPTED MANUSCRIPT

∂ 2 vij ∂ 2ϕi j 1 1 ≈ vij+1 − vij+−11 − vij−1 + vij−−11 ) , ≈ ϕi j+1 − ϕi +j −11 − ϕi −j 1 + ϕi −j −11 ) , ( ( ∂x∂t 2τ h ∂x∂t 2τ h 3 j ∂ vi 1 ≈ v j +1 − vij+−11 − 2vij +1 + 2vij −1 + vij−+11 − vij−−11 ) , 2 2 ( i +1 ∂x ∂t 2τ h ∂ 4 vij 1 ≈ ( vij++21 − vij+−21 − 2vij++11 + 2vij+−11 + 2vij−+11 − 2vij−−11 − vij−+21 + vij−−21 ) 3 ∂x ∂t 4τ h3

(28)

(7), a following set of algebraic equations are obtained 1

τ2

(v

j +1

i

− 2vij + vij −1 ) +

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where vji and φji represent v(x,t) and φ(x,t) at (xi,tj). By substituting Eqs. (26), (27) and (28) into Eq.

γ κγ 2 − 1 j k vij+1 − vij+−11 − vij−1 + vij−−11 ) + vi +1 − 2vij + vij−1 ) + 1 (ϕi j+1 − ϕi −j 1 ) ( ( 2 2τ h h 2h

  kN2 vˆ, xx ( ih ) k N2 H1 k N2 1 j j j 2 v − 2 v + v k + + k H + H ( H1 + H 2 ) =0, ( )  i +1 i i −1 1 N 2 3− h2 2 2 2   k2 j +1 kγ ϕ − 2ϕi j + ϕi j −1 ) + 2 (ϕi +j 1 − ϕi +j −11 − ϕi j−1 + ϕi j−−11 ) 2 ( i 2τ h τ 2 2 ( k2γ − kf ) ϕ j − 2ϕ j + ϕ j + k ϕ j − k1 v j − v j = 0 + ( i+1 i i−1 ) 1 i 2h ( i +1 i−1 ) h2

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SC

−

where 1

H1 = ∫ v,2x dx = 0

1

N −1 1 2 2 2 A 2π2 1  j v1 − v−j1 ) + 2∑ ( vij+1 − vij−1 ) + ( vNj +1 − vNj −1 )  , H 3 = ∫ vˆ,2x dx = s , (  0 8h  2 i =1  N −1 As π  j  v1 − v−j1 ) cos ( 0 ) + 2∑ ( vij+1 − vij−1 )cos (π xi ) + ( vNj +1 − vNj −1 ) cos ( π )  (  4  i =1 

(30)

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H 2 = ∫ v, x vˆ, x dx = 0

(29)

For calculating the time histories, initial conditions are needed. In this section, initial conditions are set as

EP

v ( x, 0 ) = A0 sin ( πx ) , ϕ ( x, 0 ) = B0 cos ( πx ) , v,t ( x, 0 ) = 0, ϕ ,t ( x, 0 ) = 0

(31)

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where A0 and B0 represent the initial vibration amplitude. Then the grid values vji and φji can be calculated by using the finite difference method. In the following calculations, h=0.01 (l=100),

τ=0.00001, A0=0.0001, and B0=0.0002. The time history of the middle point of the transporting Timoshenko beam is shown Fig. 7 with kf=0.9149, kN =26.4102, k1=207.5893, k2=0.0012,γ=4, and T=5/τ. Fig. 7 demonstrates that the first-order buckled states of the super-critical transporting Timoshenko beam, defined by Eq. (6), can be considered to be stable. The practical importance of the DFT has been well recognized. For

ACCEPTED MANUSCRIPT applying the DFT to transform the time history to the frequency domain, the sampling interval is chosen as ∆t=0.001 from the time history shown in Fig. 7. Therefore, the total number of samples is M=5/∆t. The following function is defined for the discrete Fourier transformation

1 M

M −1

∑v m =0

5m Mτ l 2

e − iω m∆t , m = 0,1, 2, ⋯ , M − 1

(32)

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F (ω ) =

where ω = 2π k ( M ∆t ) , k = 0,1, 2⋯ M − 1 . Fig. 8 shows the relationship between ω and F(ω). The first two natural frequencies of the axially transporting Timoshenko beam are clearly displayed in

SC

Fig. 8. Therefore, for any given system parameters, natural frequencies of the super-critical

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transporting beam can be determined by adopting the DFT in conjunction with the FDM.

0.0004

v(0.5,t)

0.0002

F(ω )

0.0003

-0.0002

0

1

2

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0.0000

3

t

4

0.0002

0.0000

5

0

EP

8 6

The power spectrum of Fig. 7

DFT Galerkin method

38 36 34 32

2 0

Fig. 8

40

DFT Galerkin method

4

10 15 20 25 30 35 40 45 50 ω

ω2

ω1

10

5

42

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12

ω2

0.0001

Fig. 7 The time history of the midpoint of the beam 14

ω1

30

3

4

5

γ

6

(a) The first natural frequency

7

28

3

4

5

γ

6

(b) The second natural frequency

Fig. 9. Comparisons between the DFT & FDM and the Galerkin method

7

ACCEPTED MANUSCRIPT For confirming the numerical results of the Galerkin method based on Eq. (12), the first two natural frequencies of the super-critical transporting Timoshenko beam are compared in Fig. 9. As shown in Fig. 9, the Galerkin method and the DFT predict almost the same results. Therefore, the natural frequencies of the super-critical Timoshenko beam, provided in this paper by using the two

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approaches are quite credible.

In Section 4, when utilizing the Galerkin method, natural frequencies are obtained by

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neglecting several nonlinear terms. Therefore, the effects of the nonlinear coefficient on natural frequencies cannot be fully assured by using the Galerkin method. Moreover, the Galerkin method

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cannot study the influences of the free vibration amplitude on natural frequencies. In view of the limitations of the Galerkin method, Fig. 10 presents the effects of the nonlinear coefficient and the free vibration amplitude on the first two natural frequencies of the super-critical transporting Timoshenko beam. As is seen from Fig. 10, for small-amplitude vibration, the same results are

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obtained for the different nonlinear coefficients. With increasing vibration amplitude, however, the nonlinear coefficients become more and more influential. For relatively large free vibration

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amplitude, the influence of the nonlinear coefficient on natural frequencies could be significant.

8

30.0 kN=15 kN=25 kN=40

kN=15 kN=25

29.5

kN=40

ω2

ω1

9

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10

29.0

28.5

7

0.005

0.010

0.015

0.020

0.025

0.030

A0

(a) The first natural frequency

28.0

0.005

0.010

0.015 0.020 A0

0.025

(b) The second natural frequency

Fig. 10. Effects of the nonlinear coefficient and the initial vibration amplitude

0.030

ACCEPTED MANUSCRIPT 6. The Comparison with Euler-Bernoulli Beam Theory Neglecting the effects of the shear and the angle of rotation of the cross-section from the transporting Timoshenko beam model, the governing equation of super-critical transporting

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Euler-Bernoulli beam with the simply supported ends can also be derived by introducing a coordinate transformation [Ding and Chen, 2011]

v,tt +2γ v, xt + (κγ 2 − 1) v, xx + kf2 v, xxxx =

(33)

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1 1 k N2 k2 v, xx ∫ ( v, x 2 +vˆ,2x +2v, x vˆ, x )dx + N vˆ, xx ∫ ( v, x 2 +2v, x vˆ, x )dx 2 2 0 0

vˆ ( x ) = ±

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where the buckling configurations to the first critical speed are defined as 2 2 κγ 2 − 1 − ( πkf ) sin ( πx ) kN π

(34)

By considering buckling configurations and its various spatial derivatives as constant coefficients,

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the partially linearized equation is derived as

1

1

k N2 v,tt +2γ v, xt + (κγ − 1) v, xx + k v, xxxx = v, xx ∫ vˆ,2x dx +k N2 vˆ, xx ∫ v, x vˆ, x dx 2 0 0 2

2 f

(35)

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Based on this quasi-linearization equation, natural frequencies of the super-critical vibration of the

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transporting Euler-Bernoulli beam also can be extracted by using a similar Galerkin truncation procedure.

Based on the four terms Galerkin truncation, the natural frequencies of two kinds of beam theories are compared in Fig. 11. The comparison in Fig. 11(b) illustrates that the second natural frequency of the super-critical vibration from the transporting Timoshenko beam theory is much smaller. As shown in Figs. 11(a) and 11(c), the fundamental frequency from the transporting Euler-Bernoulli beam is smaller than that of the Timoshenko beam theory near the critical speed. This is because the first critical speed of the transporting Timoshenko beam is smaller. However,

ACCEPTED MANUSCRIPT Figs. 11(a) and 11(d) demonstrate that the fundamental frequency of the Timoshenko beam becomes smaller with further increasing axial speed. Moreover, the differences of the first natural frequency between two beam theories increase with the translating speed. 40 4-term Galerkin method

38

12

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15

Euler-Bernoulli Beam Timoshenko Beam

36

ω2

9

ω1

34

Euler-Bernoulli Beam Timoshenko Beam

32

3

30

0 2.5

28 2.5

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6

3.0

3.5

4.0 4.5

γ

5.0

5.5

6.0

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4-term Galerkin method

6.5

7.0

(a) The first natural frequency 8

3.5

4.0 4.5

γ

5.0

5.5

6.0

6.5

7.0

(b) The second natural frequency

16 15

7 4-term Galerkin method

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13

ω1

5

4-term Galerkin method

14

6

ω1

4 3

1 3.0

3.1

3.2

γ

11 10 9

3.3

7 3.5

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2.9

12

Euler-Bern oulli Beam Tim osh enko Beam

EP

2

0 2.8

3.0

3.4

3.5

3.6

Euler-Bernoulli Beam Timoshenko Beam

8

3.7

(c) Local magnification of (a): near critical speed

4.0

4.5

5.0

γ

5.5

6.0

6.5

7.0

(d) Local magnification of (a): relatively high

speed

Fig. 11. Comparisons between the Euler-Bernoulli model and the Timoshenko model by using the Galerkin method The governing equation of the super-critical transporting Euler-Bernoulli beam also can be directly numerically solved by the finite difference schemes (26), (27) and (28). The time history of the middle point of the transporting Euler-Bernoulli beam is shown in Fig. 12. Similarly, natural

ACCEPTED MANUSCRIPT frequencies can be abstracted from the time history in Fig. 12. Similarly, Fig. 12 illustrates that the first-order buckled state of the super-critical transporting Euler-Bernoulli beam, defined by Eq. (34), is stable. Fig. 13 describes the first two natural frequencies obtained by the DFT. Based on the DFT & FDM, comparisons of natural frequencies of two beam theories are presented in Fig. 14. The

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same information can be drawn from Fig. 14. For the super-critical transporting continuum, there is obvious distinction among natural frequencies of the Timoshenko model and the Euler-Bernoulli

0.0002

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model.

ω1

v(0.5,t)

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0.0008

F(ω )

0.0006

0.0000

0.0004

ω2

-0.0002

0

1

2

t

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0.0002

3

4

0.0000

5

15

9

6

10 15 20 25 30 35 40 45 50 ω

Fig. 13

The power spectrum of Fig. 12

38 36

Euler-Bernoulli Beam Timoshenko Beam

ω2

ω1

12

5

40

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DFT & FDM

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Fig. 12 The time history of the midpoint of beam

0

34

Euler-Bernoulli Beam Timoshenko Beam

32 30 DFT & FDM

3 3.0

3.5

4.0

4.5

5.0

γ

5.5

6.0

6.5

7.0

(a) The first natural frequency

28 3.0

3.5

4.0

4.5

5.0

γ

5.5

6.0

6.5

7.0

(b) The second natural frequency

Fig. 14. Comparisons between the Euler-Bernoulli model and the Timoshenko model by using the DFT & FDM

ACCEPTED MANUSCRIPT 7. Natural Frequencies of the Fixed Ends In this section, the dimensionless forms fixed supported at both ends are considered as

v ( 0, t ) = 0, v (1, t ) = 0, ϕ ( 0, t ) = 0, ϕ (1, t ) = 0

(36)

the transporting Timoshenko beam are deduced as [Ding et al. 2016]

− 1) −

4k1π 2 ( kf2 − k2γ 2 )

k1 + 4π ( k − k2γ 2

2 f

(κγ

2k1

kN  k1 + 4π 2 ( kf2 − k2γ 2 ) 

2

2

)

sin 2 ( πx ) ,

− 1) −

4k1π 2 ( kf2 − k2γ 2 )

k1 + 4π 2 ( kf2 − k2γ 2 )

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ϕˆ ( x ) = ±

(κγ

2

SC

2 vˆ ( x ) = ± kN π

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Corresponding to the fixed ends, the non-trivial buckling configurations of the first critical speed of

sin ( 2πx )

(37)

Similarly, substituting v(x, t )→ vˆ + (x ) + v(x, t ) and Eq. (37) into Eq. (1) and partially linearizing the resulting equation yields

v,tt +2γ v, xt + {κγ − 1 − k1 − S } v, xx + k1ϕ , x = k vˆ, 2

2 N

+ xx

1

∫ v,

x

vˆ,+x dx,

0

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k2ϕ ,tt +2k2γϕ , xt + ( k2γ − k ) ϕ , xx + k1 (ϕ − v, x ) = 0, 2

2 f

S = (κγ − 1) − 4k1π ( k − k2γ 2

2

2 f

2

)

 k1 + 4π ( k − k2γ  2

2 f

2

)

(38)

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The solutions to Eq. (38) are supposed to take the following form as n

n

j =1

j =1

where

(39)

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v( x, t ) = ∑ q vj ( t ) φ j ( x ), ϕ ( x, t ) = ∑ qϕj ( t ) φ j ( x )

φ j ( x ) , j = 1, 2,... are

the trial functions. In the following investigation, the trial functions are

chosen as eigenfunctions of a stationary Timoshenko beam under the both fixed ends. Therefore,

φ j ( x ) = cos ( β j x ) − ch ( β j x ) + ζ j sin ( β j x ) − sh ( β j x )  , j = 1, 2⋯ ζ j = ( chβ j − cos β j ) ( sin β j − shβ j ) , j = 1, 2,⋯ ,

β1 = 4.7300,β 2 = 7.8532, β3 = 10.9956, β 4 = 14.1372, β5 = 17.2786, β 6 = 20.4204, β 7 = 23.5619, β8 = 26.7035,⋯

(40)

Obviously, unlike the simply supported boundary, the eigenfunctions of the stationary Timoshenko beam with the fixed boundary are the transcendental equations. If the trial function and the weight

ACCEPTED MANUSCRIPT function are taken as this set of transcendental equations, that is to say using the Galerkin truncation method, then it will produce a larger calculation error. Therefore, the weighted residual method (WRM) is applied for determining the natural frequencies of the transporting Timoshenko beam

w ( x ) = sin ( iπx ) , i = 1, 2⋯

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with fixed ends. The weight functions are chosen as (41)

It should be mentioned that the transcendental equations (40) and the weight functions (41) also satisfy orthogonality. After substituting Eq. (39) into Eq. (38), the weighted residual procedure leads

n

1

n

1

j =1

0

j =1

0

SC

to the following set of second-order ordinary differential equations

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∑ qɺɺvj ( t )∫ φ j ( x ) sin ( iπx ) dx + 2γ ∑ qɺ vj ( t )∫ φ j′ ( x ) sin ( iπx )dx

+ (κγ 2 − 1 − k1 − S ) ∑ q vj ( t ) ∫ φ j′′ ( x ) sin ( iπx ) dx + k1 ∑ qϕj ( t ) ∫ φ j′ ( x ) sin ( iπx )dx −k

n

2 N

n

1

j =1

0

1

n

1

j =1

0

∑ q ( t )∫ vˆ ∫ φ ′ ( x ) vˆ dx sin ( iπx )dx = 0, 1

v j

j =1

0

+ xx

+ x

j

0

n

1

j =1

0

n

1

j =1

0

(42)

k2 ∑ qɺɺϕj ( t ) ∫ φ j ( x ) sin ( iπx ) dx + 2k2γ ∑ qɺ ϕj ( t ) ∫ φ j′ ( x ) sin ( iπx ) dx n

j =1

n

1

j =1

0

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− ( k2γ 2 − kf2 ) ∑ qϕj ( t ) ∫ φ j′′ ( x ) sin ( iπx ) dx 1

0

n

1

j =1

0

EP

+ k1 ∑ qϕj ( t ) ∫ φ j′ ( x ) sin ( iπx ) dx − k1 ∑ q vj ( t ) ∫ φ j′ ( x ) sin ( iπx ) dx = 0 Then, the matrix-vector form, Eq. (12) is obtained with following coefficients

M v = ∫ φ j ( x ) sin ( iπx ) dx, G v = 2γ ∫ φ j′ ( x ) sin ( iπx )dx, 1

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0

1

0

1 1 1  K vv = (κγ 2 − 1 − k1 − S ) ∫ φ j′′ ( x ) sin ( iπx ) dx − k N2 ∫ vˆxx+  ∫ φ j′ ( x ) vˆx+ dx  sin ( iπx ) dx, 0 0 0 

(43)

Kϕv = k1 ∫ φ j′ ( x ) sin ( iπx ) dx 1

0

M ϕ = k2 ∫ φ j ( x ) sin ( iπx ) dx, Gϕ = 2k2γ ∫ φ j′ ( x ) sin ( iπx ) dx, K vϕ = − k1 ∫ φ j′ ( x ) sin ( iπx ) dx, 1

0

1

1

0

0

Kϕϕ = ( k2γ 2 − kf2 ) ∫ φ j′′ ( x ) sin ( iπx ) dx + k1 ∫ φ j ( x ) sin ( iπx ) dx 1

1

0

0

(44)

Similarly, the natural frequencies ω are introduced by Eq. (22). Then, ω are calculated by setting the determinant of coefficients to be zero. On the other hand, with the fixed boundary conditions (36), Eq. (7) is numerically solved by

ACCEPTED MANUSCRIPT the finite difference method. Then, the natural frequencies of the axially transporting Timoshenko beam are calculated by using the discrete Fourier transformation. The same parameters of the axially transporting beam as in Section 4 are adopted. For verifying the numerical results, the first natural frequencies of the Timoshenko beam with the fixed

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boundary conditions calculated from Eq. (12) are compared with the results of the DFT in Fig. 15. As shown in Fig. 15, the fundamental frequency based on the DFT almost coincides with those of Eq. (12) with 8-term truncation. Specially when the axial speed is not too great. Therefore, for the

SC

fixed supported ends, eight terms weighted residual method is needed for the first natural frequency

20

20 4-term WRM DFT

8-term WRM DFT

15

ω1

ω1

15

10

5

2

4

γ

6

EP

0

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10

0

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of the super-critical transporting Timoshenko beam.

(a) 4-term weighted residual method

8

5

0

0

2

4

γ

6

8

(b) 8-term weighted residual method

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Fig. 15. Comparisons the first natural frequency of the Timoshenko model between the DFT & FDM and the weighted residual method

The first natural frequency of the transporting beam with the simply supported boundary conditions and the fixed ends are compared in Fig. 16. For a relatively small speed, the first natural frequency of the beam with fixed ends is about twice as much as those with the simply supported ends. Interestingly, in a wide speed range, the first natural frequency of the transporting beam with the simply supported boundary conditions is larger than those with the fixed ends. On the whole, the natural frequency of the transporting beam is very sensitive to the boundary conditions.

ACCEPTED MANUSCRIPT 20

20 DFT

8-term WRM Simply Supported Fixed ends

Simply Supported Fixed Ends

15

10

10

5

5

0

0

2

4

0

γ

6

8

0

2

4

γ

6

8

(b) The DFT & FDM

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(a) The 8-term weighted residual method

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ω1

ω1

15

Fig. 16. Comparisons between the simply supported boundary conditions versus the fixed ends of

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the Timoshenko model

Fig. 17 presents the effects of bending stiffness and axial speed of the transporting beam on the first natural frequencies. Compared to the numerical results in Figs. 3 and 4, the same

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conclusions are drawn. The first natural frequency of the super-critical transporting Timoshenko beam may become smaller with the higher bending stiffness or the smaller axial speed. 21

kf=0.6

18

kf=0.8

EP

15

9 6 3 0

0

AC C

ω1

12

1

2

3

4

γ

5

12 10

ω1

kf=1.0

14

γ =6 γ =6.5 γ =7

8 6 4 2

6

7

8

9

0 0.25

(a) The first natural frequency versus the axial speed

0.50

kf

0.75

1.00

(b) The first natural frequency versus the

bending stiffness Fig. 17. Effects of the bending stiffness and the axial speed of the transporting Timoshenko beam by using the 8-term weighted residual method

ACCEPTED MANUSCRIPT

20

20

Euler-Bernoulli Beam Timoshenko Beam

Euler-Bernoulli Beam Timoshenko Beam

15

ω1

ω1

15

10

10

DFT

8-term WRM

0

0

1

2

3

4

0

γ

5

6

7

8

2

4

γ

6

8

(b) The DFT & FDM

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(a) The 8-term weighted residual method

0

RI PT

5

5

Fig. 18. Comparisons between the Euler-Bernoulli model and the Timoshenko model:

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In the sub-critical and super-critical speed range, Fig. 18 compares the first natural frequency of two kinds of beam theories. The similar conclusions are found in Fig. 18 as the comparisons for the simply supported ends shown in Fig. 11. In a certain area, the first natural frequency of the

8. Conclusions

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super-critical transporting Timoshenko beam is larger than that of the Euler-Bernoulli beam.

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The present work focuses on the free super-critical vibration of an axially transporting Timoshenko beam. Straight equilibrium of the transporting beam loses its stability at the

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super-critical speed. Based on the buckling configurations, the governing equation of the super-critical vibration of the transporting Timoshenko beam is derived. By applying the finite difference method, the time histories of the transporting beam are numerically calculated. Then natural frequencies of free vibration are extracted by utilizing the discrete Fourier transform. On the other hand, by using the convergent Galerkin method, natural frequencies of the super-critical vibration are calculated based on a partially linearized equation. Numerical simulations show that the super-critical natural frequencies are very sensitive to system parameters, including the bending stiffness, the rotary inertia and the shear force. Furthermore, the system becomes more sensitive to

ACCEPTED MANUSCRIPT the increasing axial speed. For relatively large amplitudes of the free vibration, the nonlinear coefficient could significantly affect the natural frequencies of the super-critical transporting Timoshenko beam. The numerical results are compared to those from the transporting Euler-Bernoulli beam model. The comparison demonstrates that the fundamental frequency of the

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transporting Timoshenko beam is larger than that of the Euler-Bernoulli model near the critical speed. Nevertheless, natural frequencies of the Euler-Bernoulli beam become higher with the further increase of the transporting speed. Furthermore, the difference between natural frequencies

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of two beam theories increases with the transporting speed. Natural frequency is also compared for the simply supported ends and the fixed ends. Comparisons interestingly show that the fundamental

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frequency with the simply supported ends is larger in a wide axial speed range.

Acknowledgments

The authors gratefully acknowledge the support of the National Natural Science Foundation of

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China (No. 11422214, 11372171) and the State Key Program of the National Natural Science Foundation of China (No. 11232009).

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References

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[1] An, C., Su, J., 2014. Dynamic response of axially moving Timoshenko beams: integral

transform solution. Applied Mathematics and Mechanics (English Edition). 35, 1421–1436. [2] Chang, J.R., Lin, W.J., Huang, C.J., Choi, S.T., 2010, Vibration and stability of an axially

moving Rayleigh beam. Applied Mathematical Modelling. 34, 1482–1497. [3] Chen, L.Q., 2005. Analysis and control of transverse vibrations of axially moving strings.

Applied Mechanics Reviews. 58, 91–116. [4] Chen, L.Q., Tang, Y.Q., Lim, C.W., 2010. Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams. Journal of Sound and Vibration. 329, 547–565.

ACCEPTED MANUSCRIPT [5] Chen, L.Q., Yang, X.D., 2006. Vibration and stability of an axially moving viscoelastic beam with hybrid supports. European Journal of Mechanics a-Solids. 25, 996-1008. [6] Chen, L.Q., Wang, B., 2009. Stability of axially accelerating viscoelastic beams: asymptotic perturbation analysis and differential quadrature validation. European Journal of Mechanics

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a-Solids. 28, 786-791.

[7] Chen, S.H., Huang, J.L., Sze, K.Y., 2007. Multidimensional Lindstedt-Poincare method for

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nonlinear vibration of axially moving beams. Journal of Sound and Vibration. 306, 1–11. [8] Cojocaru, E.C., Irschik, H., Schlacher, K., 2003. Concentrations of pressure between an

Mechanics-ASCE. 129,1076–1082.

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elastically supported beam and a moving Timoshenko-beam. Journal of Engineering

[9] Ding, H., Chen, L.Q., 2008. Stability of axially accelerating viscoelastic beams: multi-scale

analysis with numerical confirmations. European Journal of Mechanics A/Solid. 27,

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Highlights:

1) Timoshenko model is used to nonlinear vibration of super-critical transporting beams. 2) Natural frequencies are extracted by two different numerical approaches.

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3) Transporting Timoshenko beam may offer a higher super-critical fundamental frequency.

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4) Fundamental frequency with simply supported ends may larger than those with fixed ends.