Stability of axially accelerating viscoelastic beams: multi-scale analysis with numerical confirmations

European Journal of Mechanics A/Solids 27 (2008) 1108–1120

Stability of axially accelerating viscoelastic beams: multi-scale analysis with numerical confirmations Hu Ding a , Li-Qun Chen a,b,∗ a Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200444, China b Department of Mechanics, Shanghai University, Shanghai 200444, China

Received 7 May 2007; accepted 20 November 2007 Available online 7 February 2008

Abstract Stability is investigated for an axially accelerating viscoelastic beam. The material time derivative is used in the viscoelastic constitutive relation, not simply the partial time derivative. The method of multiple scales is applied directly to the governing equation without discretization. When the axial speed is characterized as a simple harmonic variation about the constant mean speed, the instability conditions are presented for axially accelerating viscoelastic beams constrained by simple supports with rotational springs in parametric resonance. The finite difference schemes are developed to solve numerically the equation of axially accelerating viscoelastic beams with fixed supports for the instability regions in the principal parametric resonance. The numerical calculations confirm the analytical results. Numerical examples show the effects of the constraint stiffness, the mean axial speed, and the viscoelasticity. © 2008 Published by Elsevier Masson SAS. Keywords: Axially accelerating beam; Parametric resonance; Instability; The material time derivative; Method of multiple scales; The finite difference

1. Introduction Axially moving beams can represent many engineering devices (Wickert and Mote, 1988; Abrate, 1992). Under certain conditions, large transverse vibration occurs due to parametric vibration resulted from the variation of the beam tension or the beam axial speed. Transverse parametric vibration of axially accelerating elastic beams has been extensively analyzed. Although Pasin (1972) first studied the problem as early as in 1972, much progress was achieved recently. Öz, Pakdemirli and Özkaya (1998) employed the method of multiple scales to study dynamic stability of an axially accelerating beam with small bending stiffness. Özkaya and Pakdemirli (2000) combined the method of multiple scales and the method of matched asymptotic expansions to construct non-resonant boundary layer solutions for an axially accelerating beam with small bending stiffness. Öz and Pakdemirli (1999) and Öz (2001) applied the method of multiple scales to calculate analytically the stability boundaries of an axially accelerating beam under pinned-pinned and clamped* Corresponding author.

E-mail address: [email protected] (L.-Q. Chen). 0997-7538/$ – see front matter © 2008 Published by Elsevier Masson SAS. doi:10.1016/j.euromechsol.2007.11.014

H. Ding, L.-Q. Chen / European Journal of Mechanics A/Solids 27 (2008) 1108–1120

1109

clamped conditions respectively. Parker and Lin (2001) adopted a 1-term Galerkin discretization and the perturbation method to study dynamic stability of an axially accelerating beam subjected to a tension fluctuation. Özkaya and Öz (2002) used an artificial neural network algorithm to determine stability boundary of an axially accelerating beam. Suweken and Horssen (2003) applied the method of multiple scales to a discretized system via the Galerkin method to study the dynamic stability of an axially accelerating beam with pinned-pinned ends. In addition to elastic beams, axially accelerating viscoelastic beams have recently been investigated. Chen, Yang and Cheng (2004) applied the averaging method to a discretized system via the Galerkin method to present analytically the stability boundaries of axially accelerating viscoelastic beams with clamped-clamped ends. Chen and Yang (2005) applied the method of multiple scales without discretization to obtain analytically the stability boundaries of axially accelerating viscoelastic beams with pinned-pinned or clamped-clamped ends. Yang and Chen (2006) applied the method of multiple scales to present analytically vibration and stability of an axially moving beam constituted by the viscoelastic constitutive law of an integral type. Chen and Yang (2006) applied the method of multiple scales to present analytically vibration and stability of an axially moving beam constrained by simple supports with rotational springs. The Kelvin model containing the partial time derivative was used to describe the viscoelastic behavior of beam materials. Mockensturm and Guo (2005) convincingly argued that the Kelvin model generalized to axially moving materials should contain the material time derivative to account for the energy dissipation in steady motion. Actually, the material time derivative was also employed in the Kelvin model of axially moving materials by Marynowski and Kapitaniak (2002), Marynowski (2004), Yang and Chen (2005) and Marynowski (2006), as well as in the threeparameter viscoelastic model by Marynowski and Kapitaniak (2007). However, there is no investigation on stability of axially accelerating beams constituted by a viscoelastic constitutive law containing the material time derivative. In all available studies via the method of multiple scales, it was assumed that only two vibration modes with corresponding natural frequencies involved in summation resonance contribute to dynamic behaviors. The effects of other modes have not been mathematically treated, although they were physically assumed to be neglectable. The modes not involved in summation resonance are also considered in the present investigation. In spite of the fact that there have been many approximately analytical investigations on stability of axially accelerating beams, there are very limited researches on the topic via the direct numerical approaches such as finite difference schemes. Consequently, the results obtained via approximately analytical approaches such as the method of multiple scales cannot be numerically conformed. To address the lacks of research in these aspects, the present investigation studies the stability in parametric resonance of an axially accelerating viscoelastic beam based on the numerical solutions via the finite difference approach. In the preset paper, the authors revisit the problem addressed (Chen and Yang, 2006) with three main improvements. In the modeling, the material time derivative is used in the Kelvin model. In the analysis via the method of multiple scales, the modes not involved in summation resonance are taken into consideration, and are demonstrated to have little effects on the stability. The finite difference scheme is developed to solve numerically the governing equation, and the numerical results confirm the stability condition derived from the method of multiple scales. 2. The governing equation A uniform axially moving viscoelastic beam, with density ρ, cross-sectional area A, moment of inertial I and initial tension P0 , travels at time-dependent axial transport speed v(T ) between two motionless ends separated by distance L. Consider only the bending vibration described by the transverse displacement U (X, T ), where T is the time and X is the axial coordinate. The Newton second law of motion yields  2 2  dv ∂U ∂ U ∂ 2 U (X, T ) ∂ 2 M(X, T ) ∂ 2U 2∂ U = P0 + + v ρA + 2v − , (1) 2 2 ∂X∂T dT ∂X ∂T ∂X ∂X 2 ∂X 2 where M(X, T ) is the bending moment given by  M(X, T ) = − Zσ (X, Z, T ) dA,

(2)

A

where ZX-plane is the principal plane of bending, and σ (X, Z, T ) is the disturbed normal stress. At both ends, the beam are constrained by simple supports with rotational springs whose the spring stiffness constant are K1 and K2

1110

H. Ding, L.-Q. Chen / European Journal of Mechanics A/Solids 27 (2008) 1108–1120

respectively. Nullifying the transverse displacements and balancing the bending moment at both ends lead to the boundary conditions   ∂ 2 U  ∂U  U (0, T ) = 0, EI − K1 = 0, ∂X X=0 ∂X 2 X=0   ∂ 2 U  ∂U  U (L, T ) = 0, EI + K2 = 0. (3)  2 ∂X  ∂X X=L

X=L

In the following, assume that K1 = K2 = K0 . The viscoelastic material of the beam with the constitution relation   ∂ε(X, Z, T ) ∂ε(X, Z, T ) +v , σ (X, Z, T ) = Eε(X, Z, T ) + η ∂T ∂X

(4)

where ε(X, Z, T ) is the axial strain, E is the stiffness constant, and η is the viscosity coefficient. For small deflections, the strain-displacement relation is ∂ 2 U (X, T ) . ∂X 2 Substitution of Eqs. (4) and (5) into Eq. (2) and then substitution the resulting equation into Eq. (1) lead to  2 2  dv ∂U ∂ 2U ∂ 2U ∂ 4U ∂ 5U ∂ 5U ∂ U 2∂ U − P0 + + v + 2v + EI + ηI + vηI = 0. ρA 2 2 2 4 4 ∂X∂T dT ∂X ∂T ∂X ∂X ∂X ∂T ∂X ∂X 5 ε(X, Z, T ) = −Z

Introduce the dimensionless variables and parameters   X U P0 ρA u= , x= , t =T , γ =v , 2 L L P0 ρAL

vf2 =

EI , P0 L2

εα =

L3

Iη , √ ρAP0

(5)

(6)

(7)

where bookkeeping device ε is a small dimensionless parameter accounting for the fact that the viscosity coefficient is very small. Eq. (6) can be cast into the dimensionless form 4 dγ ∂u ∂ 5u ∂ 2u ∂ 2u ∂ 2u ∂ 5u 2 2∂ u + + (γ + εαγ + 2γ − 1) + v + εα = 0. f ∂x∂t dt ∂x ∂t 2 ∂x 2 ∂x 4 ∂x 4 ∂t ∂x 5

(8)

3. Stability condition via the method of multiple scales: analytical procedure In the present investigation, the axial speed is assumed to be a small simple harmonic variation about the constant mean speed, γ (t) = γ0 + εγ1 sin ωt,

(9)

where γ0 is the mean axial speed, and εγ1 and ω are respectively the amplitude and the frequency of the axial speed variation, all in the dimensionless form. Substitution of Eq. (9) into Eq. (8) and neglect of higher order ε terms in the resulting equation yield ∂u ∂ 2u ∂ 2u ∂ 2u + Ku = −2εγ − 2εγ + G sin ωt γ sin ωt 1 0 1 ∂t ∂x∂t ∂t 2 ∂x 2 5 ∂ u ∂ 5u ∂u − εα 4 − εαγ0 5 , − εωγ1 cos ωt ∂x ∂x ∂t ∂x where the mass, gyroscopic, and linear stiffness operators are respectively defined as M

(10)

∂ ∂4 ∂2 , K = (γ02 − 1) 2 + vf2 4 . (11) ∂x ∂x ∂x In the following, the method of multiple scales will be employed to determine the stability boundary. In contrast to the investigation on the effects of viscoelasticity on free vibration, a first order approximation is enough to obtain the stability boundary (Chen and Yang, 2006). Suppose that the uniform approximate solution to Eq. (10) is M = I,

G = 2γ0

u(x, t; ε) = u0 (x, T0 , T1 ) + εu1 (x, T0 , T1 ) + · · · ,

(12)

H. Ding, L.-Q. Chen / European Journal of Mechanics A/Solids 27 (2008) 1108–1120

1111

where T0 = τ and T1 = ετ are respectively the fast and slow time scales. Substitution of Eq. (12) and the following relationships ∂2 ∂2 ∂2 = + 2ε + ··· ∂T0 ∂T1 ∂t 2 ∂T02

∂ ∂ ∂ = +ε + ···, ∂t ∂T0 ∂T1

(13)

into Eq. (10) and then equalization of coefficients of ε 0 and ε 1 in the resulting equation lead to M and

∂u0 ∂ 2 u0 +G + Ku0 = 0 2 ∂T0 ∂T0

(14)

  2 ∂ 2 u1 ∂ 2 u0 ∂ 2 u0 ∂u1 ∂ 2 u0 ∂ u0 M +G + Ku1 = −2 − 2γ0 − 2γ1 sin ωt + γ0 2 ∂T0 ∂T0 ∂T1 ∂x∂T1 ∂x∂T0 ∂x ∂T02 − γ1 ω cos ωt

∂ 5 u0 ∂ 5 u0 ∂u0 −α 4 − αγ0 5 . ∂x ∂x ∂T0 ∂x

Wickert and Mote have obtained the solution to Eq. (14)

 φk (x)Ak (T1 )eiωk T0 + φ¯ k (x)A¯ k (T1 )e−iωk T0 u0 (x, T0 , T1 ) =

(15)

(16)

k=0,1,...

where the over bar denotes complex conjugation, and the kth natural frequency and the kth complex eigenfunction can be determined by the boundary conditions. If the axial speed variation frequency ω approaches the sum of any two natural frequencies of the system (14), the summation parametric resonance may occur. A detuning parameter σ is introduced to quantify the deviation of ω from ωm + ωn (m  n), and ω is described by ω = ωm + ωn + εσ,

(17)

where ωk (k = m, n) is the kth frequency of free vibration described by Eq. (14). To investigate the summation parametric response with the possible contributions of modes not involved the resonance, the solution to Eq. (15) be expressed as u0 (x, T0 , T1 ) = φl (x)Al (T1 )eiωl T0 + φm (x)Am (T1 )eiωm T0 + φn (x)An (T1 )eiωn T0 + φo (x)Ao (T1 )eiωo T0 + cc,

(18)

where l < m, n < 0, and cc stands for the complex conjugate of all preceding terms on the right hand of an equation. Substituting Eqs. (17) and (18) into Eq. (15), expressing the trigonometric functions in the exponential form, and regrouping all terms finally yield 4 ∂ 2 u1 ∂ 2 u1 ∂ 2 u1 2 2 ∂ u1 + 2γ + (γ − 1) + v 0 0 f ∂x∂T0 ∂x 2 ∂x 4 ∂T02  

1  T1 eiωm T0 = −2A˙ m (iωm φm + γ0 φm ) + γ1 (ωn − ωm )φ¯ n + iγ0 φ¯ n φ¯ n eiσ m 2  

1   ¯ iσ T1 φm en eiωn T0 + −2A˙ n (iωn φn + γ0 φn ) + γ1 (ωm − ωn )φ¯ m + iγ0 φ¯ m 2 

+ −2A˙ l (iωl φl + γ0 φl ) − iαωl Al φl(4) + αγ0 Al φl(5) eiωl T0 

+ −2A˙ 0 (iω0 φ0 + γ0 φ0 ) − iαω0 A0 φ0(4) + αγ0 Ao φo(5) eiωo T0



  (4) (5) iωm T0 − iαωm Am φm + αγ0 Am φm − iαωn An φn(4) + αγ0 An φn(5) eiωn T0 + cc + NST, e

(19)

where the dot and the prime denote derivation with respect to the slow time variable T1 and the dimensionless spatial variable x respectively, and NST stands for the terms that will not bring secular terms into the solution. Eq. (19) has

1112

H. Ding, L.-Q. Chen / European Journal of Mechanics A/Solids 27 (2008) 1108–1120

a bounded solution only if a solvability condition holds. The solvability condition demands the following orthogonal relationships (Chen and Zu, 2008)   −2A˙ l (iωl φl + γ0 φl ) − iαωl Al φl(4) − αγ0 Al φl(5) , φl = 0,     1    ¯ iσ T1 (4) (5) ˙ ¯ ¯ −2An (iωn φn + γ0 φn ) + γ1 (ωm − ωn )φm + iγ0 φm φm e − iαωn An φn − αγ0 An φn , φn = 0, 2     1    ¯ iσ T1 (4) (5) ˙ ¯ ¯ −2Am (iωm φm + γ0 φm ) + γ1 (ωn − ωm )φn + iγ0 φn φn e − iαωm Am φm − αγ0 Am φm , φm = 0, 2   −2A˙ 0 (iωo φo + γ0 φo ) − iαωo Ao φo(4) − αγ0 Ao φo(5) , φo = 0, (20) where the inner product is defined for complex functions on [0, 1] as 1 f, g =

f g¯ dx.

(21)

0

Application of the distributive law of the inner product to Eq. (20) leads to A˙ l + αcll Al = 0, A˙ m + αcmm Am + γ1 dmn A¯ n eiσ T1 = 0, A˙ n + αcnn An + γ1 dnm A¯ m eiσ T1 = 0,

(24)

A˙ o + αcoo Ao = 0,

(25)

where

 1 (5) (4) φk φ¯ k dx + γ0 0 φk φ¯ k dx (k = l, m, n, o), 1 1 2(iωk 0 φk φ¯ k dx + γ0 0 φk φ¯ k dx)  1  1  (ωm − ωn ) 0 φ¯ m φ¯ n dx + 2iγ0 0 φ¯ m φ¯ n dx dnm = − , 1 1 4(iωn 0 φn φ¯ n dx + γ0 0 φn φ¯ n dx) 1 1 (ωn − ωm ) 0 φ¯ n φ¯ m dx + 2iγ0 0 φ¯ n φ¯ m dx dmn = − . 1 1  φ¯ dx) 4(iωm 0 φm φ¯ m dx + γ0 0 φm m ckk =

iωk

(22) (23)

1 0

(26)

(27)

These coefficients can be determined by the natural frequencies and the modal functions calculated from Eq. (14) with the boundary conditions, and are independent of the parametric excitation due to the variation of axial speed. Under all boundary conditions concerned here, it is numerically demonstrated that ckk is a positive real number for beams moving with a subcritical axial speed, namely, ckk = Re(ckk ) > 0 and Im(ckk ) = 0. For positive real number cll and coo , the solutions to Eqs. (22) and (25) decay to zero exponentially. In addition, Eqs. (22) and (25) do not coupled with Eqs. (23) and (24). Therefore, the lth and the oth modes have actually no effect on the stability. Form Eqs. (23) and (24), one can derive the stability boundary in the same form as equation (50) in Chen and Yang (2006), although the coefficients are different. Eq. (57) in Chen and Yang (2006) gives the analytical expression of the stability boundary in summation parametric resonance  2  2  σ σ (28) (cnn − cmm ) + (cnn + cmm )2 + α 2 cnn cmm + γ12 dnm d¯mn = 0. 2 4 Eq. (28) is the analytical expression of the stability boundary in summation parametric resonance. Therefore, the instability region is given as   γ12 Re(dnm d¯mn ) − α 2 cnn cmm γ12 Re(dnm d¯mn ) − α 2 cnn cmm −2 <σ < 2 , (29) 1 + κ2 1 + κ2 where κ=

cmm − cnn . cmm + cnn

(30)

H. Ding, L.-Q. Chen / European Journal of Mechanics A/Solids 27 (2008) 1108–1120

1113

Now we consider the principal parametric resonance where the variation frequency ω approaches two times of a natural frequency of the system (10). Denote ω = 2ωk + εσ.

(31)

Let m = n in Eq. (29), then the resulting equation gives the stability boundary in kth principal parametric resonance.   R 2 < σ < 2 γ 2 |d |2 − α 2 cR 2 , −2 γ12 |dkk |2 − α 2 ckk (32) kk 1 kk where

1 φk φ¯ k dx + γ0 0 φ  k φ¯ k dx ckk = , 1 1 2(iωk 0 φk φ¯ k dx + γ0 0 φk φ¯ k dx) iωk

1 0

dkk = −

2(iωk

1 0

1

φ¯ k φ¯ k dx . 1 φk φ¯ k dx + γ0 0 φk φ¯ k dx) iγ0

0

(33)

4. Stability condition via the method of multiple scales: numerical demonstrations The boundary conditions of an axially moving beam with hybrid supports in dimensionless form are     ∂u  ∂u  ∂ 2 u  ∂ 2 u  − k = 0, u(1, t) = 0, + k = 0, u(0, t) = 0, 0 0 ∂x x=0 ∂x x=1 ∂x 2 x=0 ∂x 2 x=1

(34)

where K0 . (35) EI L Under the boundary conditions (34), the eigenfunction corresponding to the nth natural frequency ωn is (Chen and Yang, 2006) k0 =

φn (x)

ik(eiβ1n + eiβ3n )(β1n − β3n ) + (eiβ1n − eiβ3n )[k 2 + (β1n + β4n )(β3n + β4n )] (β1n − β4n ) = c1 eiβ1n x − ik(eiβ2n + eiβ3n )(β2n − β3n ) + (eiβ2n − eiβ3n )[k 2 + (β2n + β4n )(β3n + β4n )] (β2n − β4n ) ik(eiβ1n + eiβ2n )(β1n − β2n ) + (eiβ1n − eiβ2n )[k 2 + (β1n + β4n )(β2n + β4n )] (β1n − β4n ) iβ3n x e × eiβ2n x − ik(eiβ2n + eiβ3n )(β2n − β3n ) + (eiβ2n − eiβ3n )[k 2 + (β3n + β4n )(β2n + β4n )] (β3n − β4n )  ik(eiβ1n + eiβ3n )(β1n − β3n ) + (eiβ1n − eiβ3n )[k 2 + (β1n + β4n )(β3n + β4n )] (β1n − β4n ) + −1 + ik(eiβ2n + eiβ3n )(β2n − β3n ) + (eiβ2n − eiβ3n )[k 2 + (β2n + β4n )(β3n + β4n )] (β2n − β4n )

 ik(eiβ1n + eiβ2n )(β1n − β2n ) + (eiβ1n − eiβ2n )[k 2 + (β1n + β4n )(β2n + β4n )] (β1n − β4n ) iβ4n x , (36) e + ik(eiβ2n + eiβ3n )(β2n − β3n ) + (eiβ2n − eiβ3n )[k 2 + (β3n + β4n )(β2n + β4n )] (β3n − β4n )

where βj n (j = 1, 2, 3, 4; n = 1, 2, . . .) are eigenvalues of simple supported with rotational springs case. Let the stiffness of the spring k → ∞ in Eq. (36), one can get the modal function of the axially moving beam on fixed supports (Öz, 2001). (β4n − β1n )(eiβ3n − eiβ1n ) iβ2n x (β4n − β1n )(eiβ3n − eiβ1n ) iβ3n x e e − φn (x) = c1 eiβ1n x − (β4n − β2n )(eiβ3n − eiβ2n ) (β4n − β3n )(eiβ3n − eiβ3n )

  (β4n − β1n )(eiβ3n − eiβ1n ) (β4n − β1n )(eiβ2n − eiβ1n ) iβ4n x (37) − e − 1− (β4n − β2n )(eiβ3n − eiβ2n ) (β4n − β3n )(eiβ2n − eiβ3n ) The ωn and βj n (j = 1, 2, 3, 4; n = 1, 2, . . .) can be calculated by numerically (Chen and Yang, 2005). Numerical examples are presented to demonstrate the effects of the constraint stiffness, the mean axial speed and the beam viscoelasticity on the instability boundary in summation parametric resonance. Fig. 1 shows the effects of the constraint stiffness for γ0 = 2.0, α = 0.0001 and k0 = 1.0 (solid line), 2.0 (dashed line), 3.0 (dash-dotted line). Fig. 2 illustrates the effect of the mean axial speed for α = 0.0001, k0 = 2.0, γ0 = 1.0 (solid line), 2.0 (dashed line), 3.0 (dash-dotted line). Fig. 3 depicts the effect of the beam viscoelasticity for k0 = 2.0, γ0 = 2.0, α = 0 (solid line), 0.0001 (dashed line), 0.0002 (dash-dotted line). The larger constraint stiffness, the smaller mean axial speed, and the larger viscosity coefficient lead to the larger instability threshold of γ1 for given σ , and the smaller instability range

1114

H. Ding, L.-Q. Chen / European Journal of Mechanics A/Solids 27 (2008) 1108–1120

Fig. 1. The effects of the constraint stiffness on stability boundaries in summation parametric resonance.

Fig. 2. The effects of the mean axial speed on stability boundaries in summation parametric resonance.

Fig. 3. The effects of the beam viscoelasticity on stability boundaries in summation parametric resonance.

of σ for given γ1 . That is, the increasing constraint stiffness, the decreasing mean axial speed, and the increasing viscosity coefficient make the instability boundaries move towards the increasing direction of γ1 in plane (ω, γ1 ) and the instability regions become narrow. Figs. 4, 5 and 6 demonstrate respectively the effects of the constraint stiffness, the mean axial speed and the beam viscoelasticity on the instability boundary in principal parametric resonance. In Fig. 4, γ0 = 2.0, α = 0.0001 and k0 = 1.0 (solid line), 2.0 (dashed line), 3.0 (dash-dotted line). In Fig. 5, α = 0.0001, k0 = 2.0, γ0 = 1.0 (solid line), 2.0 (dashed line), 3.0 (dash-dotted line). In Fig. 6, k0 = 2.0, γ0 = 2.0, α = 0 (solid line), 0.0001 (dashed line), 0.0002 (dash-dotted line). Similar to the trends in summation parametric resonance, the larger constraint stiffness, the smaller mean axial speed, and the larger viscosity coefficient lead to the larger instability threshold of γ1 for given σ , and the smaller instability range of σ for given γ1 . In addition, the instability regions in the first principal parametric resonance are larger than those in the second principal parametric resonance. The instability boundaries in the second principal parametric resonance are more sensitive to the viscosity coefficient. Therefore, it can be inferred that the instability occurs more possibly in low-order principal parametric resonance, and the viscosity coefficient effect more on the high-order principal parametric resonance. 5. Numerical investigations via the finite difference scheme The finite difference method will be employed to solve numerically equation 4 dγ ∂u ∂ 5u ∂ 2u ∂ 2u ∂ 5u ∂ 2u 2 2∂ u + + (γ + αγ + 2γ − 1) + v + α = 0. f ∂x∂t dt ∂x ∂t 2 ∂x 2 ∂x 4 ∂x 4 ∂t ∂x 5

(38)

H. Ding, L.-Q. Chen / European Journal of Mechanics A/Solids 27 (2008) 1108–1120

(a) The first principal resonance

1115

(b) The second principal resonance

Fig. 4. The effect of the constraint stiffness on instability boundaries in principal parametric resonance.

(a) The first principal resonance

(b) The second principal resonance

Fig. 5. The effect of the mean axial speed on instability boundaries in principal parametric resonance.

(a) The first principal resonance

(b) The second principal resonance

Fig. 6. The effect of the beam viscoelasticity on instability boundaries in principal parametric resonance.

Eq. (38) is the same as Eq. (8) with the exception that ε = 1 here. Other numerical methods such as the Galerkin ˇ finite-element method (Cepon and Boltežar, 2007) may also serve the propose. For an axially moving beam with fixed supports, the boundary conditions in dimensionless form are   ∂u  ∂u  = = 0. (39) u(0, t) = u(1, t) = 0, ∂x  ∂x  x=0

x=1

1116

H. Ding, L.-Q. Chen / European Journal of Mechanics A/Solids 27 (2008) 1108–1120

With the emphasis on transverse vibration, the initial conditions for Eq. (38) are u,t (x, 0) = 0,

u(x, 0) = Dx(1 − x),

(40)

where D determines the initial amplitude of vibration. Introduce the L × T equispaced mesh grid with time step τ and space step h xj = j h (j = 0, 1, 2, . . . , L, h = l/L),

(41)

tn = nτ

(42)

(n = 0, 1, 2, . . . , T ).

Denote the function values u(x, t) at (xj , tn ) as unj . Application of centered difference approximations (Boresi et al., 2003) to the time, space and mixed partial derivatives leads to u,tt =

− 2unj + un−1 un+1 j j

u,x =

unj−2 − 8unj−1 + 8unj+1 − unj+2

, 12h τ2 −unj−2 + 16unj−1 − 30unj + 16unj+1 − unj+2 u,xx = , 12h2 n−1 n−1 n−1 −unj+2 + 8unj+1 − 8unj−1 + unj−2 + un−1 j +2 − 8uj +1 + 8uj −1 − uj −2 , u,xt = 12hτ n n n n n uj −2 − 4uj −1 + 6uj − 4uj +1 + uj +2 u,xxxx = , h4 n−1 n−1 n−1 + 4un−1 unj+2 − 4unj+1 + 6unj − 4unj−1 + unj−2 − un−1 j +2 + 4uj +1 − 6uj j −1 − uj −2 u,xxxxt = , τ h4 −unj−3 + 4unj−2 − 5unj−1 + 5unj+1 − 4unj+2 + unj+3 . (43) u,xxxxx = 2h5 As the time step is much smaller than the space step, all space partial derivatives are up to the fourth order, while all time partial derivatives are only up to the second order. Substitution of Eq. (43) into Eq. (38) leads to a set of algebraic equations with respect to unj that can be solved as under the boundary conditions (39) and the initial conditions (40). Then the resulting grid values unj are used in the finite difference schemes as an approximation to the continuous solutions u(x, t) to Eq. (38). To obtain the stability boundary, the following indexes are introduced σ(1) =

L 0.6T j =0 n=0.4T

|vjn |,

,

σ(2) =

L 0.8T j =0 n=0.6T

|vjn |,

σ(3) =

L T

|vjn |.

(44)

j =0 n=0.8T

Therefore, σ (1) and σ (2) and σ (3) indicate the accumulative of absolute value of the particle transverse displacement governed by Eq. (38) respectively from 0.4T to 0.6T and from 0.6T to 0.8T and from 0.8T to T . If σ (2) bigger than σ (1) and σ (3) bigger than σ (2), one can consider the parametric resonance is instability. Numerical simulations indicate that the stability boundary does not depend sensitively on the time and space steps. Fig. 7 shows the effects of the time and space steps on the first two principal parametric resonance, in which vf = 0.8, γ0 = 4.0, α = 0.0001, τ = 10−4 (solid lines), 10−6 (dots), and h = 2 × 10−2 (solid lines), 10−2 (dots). The significant differences in the steps result in very close outcomes. Therefore, in the following calculations, choose τ = 10−6 , and h = 10−2 (and hence L = 100) in the finite difference scheme. Consider an axially moving beam with vf = 0.8. Figs. 8 and 9 demonstrate respectively the effects of the beam viscoelasticity and the mean axial speed on the instability boundary in principal parametric resonance. In Fig. 8(a), γ0 = 4.0, α = 0.0003 (solid line), 0.0005 (dashed line), 0.001 (dash-dotted line). In Fig. 8(b), γ0 = 4.0, α = 0.000005 (solid line), 0.0001(dashed line), 0.0002 (dash-dotted line). In Figs. 9, γ0 = 3.5 (solid line), 4.0 (dashed line), 4.5 (dash-dotted line), and α = 0.0005 in Fig. 9(a) and α = 0.0001 in Fig. 9(a). It is concluded that the stability boundaries in the second principal parametric resonance are more sensitive to the viscosity coefficient. The stability boundaries in the first principal parametric resonance are more sensitive to the mean axial speed. In addition, Figs. 8 and 9 also demonstrate the instability regions in the first principal parametric resonance are larger than those in the second principal parametric resonance. It should be noted that, shown in Fig. 9, the stability boundaries in the first and second

H. Ding, L.-Q. Chen / European Journal of Mechanics A/Solids 27 (2008) 1108–1120

(a) The first principal resonance

1117

(b) The second principal resonance

Fig. 7. The effects of the time and space steps on stability boundaries in principal parametric resonance.

(a) The first principal resonance

(b) The second principal resonance

Fig. 8. The effects of the beam viscoelasticity on stability boundaries: the finite difference.

(a) The first principal resonance

(b) The second principal resonance

Fig. 9. The effects of the mean axial speed on stability boundaries: the finite difference.

principal resonances have the opposite trend with the increase of the mean axial speed. Due to the introduction of the material time derivative into the constitutive relation, the added “steady dissipation” term (Mockensturm and Guo, 2005) increase the mean axial speed, and the dissipation enhances the beam viscoelasticity which effects more on the higher order principal resonance.

1118

H. Ding, L.-Q. Chen / European Journal of Mechanics A/Solids 27 (2008) 1108–1120

(a) The first principal resonance

(b) The second principal resonance

Fig. 10. The effect of the beam viscoelasticity on instability boundaries: the multi-scale analysis.

(a) The first principal resonance

(b) The second principal resonance

Fig. 11. The effect of the mean axial speed on stability boundaries: the multi-scale analysis.

Now the method of multiple scales will be applied to the case. Consider still an axially moving beam with vf = 0.8. For the given γ0 , for example, γ0 = 4.0, the first two natural frequencies and coefficients in corresponding eigenfunctions (37) and be numerically calculated as ω1 = 9.5146, β11 = 6.6676, β21 = −2.4953 + 2.5344i, β31 = −2.4953 − 2.5344i, β41 = −1.6771 and ω2 = 43.3456, β12 = 10.2236, β22 = −2.4997 + 6.9798i, β32 = −2.4997 − 6.9798i, β42 = −5.2241. In principal parametric resonance, equation (33) gives c11 = 605.0250, c22 = 2994.6641, d11 = 1.5272 − 0.6178i, d22 = 0.7776 − 0.7987i. The stability boundaries in the first and second principal resonance in plane σ –γ1 are illustrated respectively in Fig. 10(a) for α = 0.0001 (solid line), 0.0005 (dashed line), 0.001 (dash-dotted line) and Fig. 10(b) for α = 0.00001 (solid line), 0.0001(dashed line), 0.0002 (dash-dotted line). Fig. 11 demonstrates the effects of the mean axial speed on the stability boundaries in principal parametric resonance. In Fig. 11, γ0 = 3.5 (solid line), 4 (dashed line), 4.5 (dash-dotted line), and α = 0.0005 in Fig. 11(a) and α = 0.0001 in Fig. 11(b). Similar, the stability boundary in the first principal resonance is less sensitive to the change of the beam viscoelasticity, while the stability boundaries in the first principal parametric resonance are more sensitive to the mean axial speed, and the instability regions in the first principal parametric resonance are larger than those in the second principal parametric resonance. Especially, as shown in Fig. 11, the mean axial speed has an opposite effects on the stability boundaries in the first and the second principal resonance, which is the same as shown in Fig. 9. The numerical simulations indicate that changing trends predicted by the two methods are qualitatively same. In both cases, the increasing beam viscoelasticity and the decreasing (increasing) mean axial speed in the first (second) principal resonance make the stability boundaries move towards the increasing direction of γ1 in plane (ω, γ1 ) and the instability regions become narrow. However, there are slightly quantitative differences. In fact, the contrasts are made in Fig. 12 in which vf = 0.8 and γ0 = 4.0 and α = 0.001, 0.0002, 0.0005, 0.0001. In these figures, the solid and dot lines represent the results of the method of finite difference and the method of multiple scales respectively. In most cases, the instability regions computed via the method of multiple scales is smaller.

H. Ding, L.-Q. Chen / European Journal of Mechanics A/Solids 27 (2008) 1108–1120

(a) The first principal resonance (α = 0.001)

(b) The second principal resonance (α = 0.0002)

(c) The first principal resonance (α = 0.0005)

(d) The second principal resonance (α = 0.0001)

1119

Fig. 12. Comparison of two methods.

6. Conclusions This paper is devoted to parametric vibration of an axially accelerating beam. The beam is constituted by the Kelvin model using the material time derivative, is constrained by simple supports with rotational springs at both ends, and moves at an axial speed fluctuating harmonically about a constant mean speed. In principal and summation parametric resonance, the method of multiple scales is applied to demonstrate the possibility of instability and to establish condition of the stability. The case with fixed supports is numerically investigated the method of finite difference respectively for the instability regions in the principal parametric resonance. The analytical results are confirmed by the numerical calculations. Both the analytical operations and the numerical evaluations demonstrate the following trends. (1) With the increase beam viscoelasticity and the constraint stiffness, the instability regions move up and become narrow in both summation and principal parametric resonance. (2) When the material time derivative is used in the constitutive relation, the increasing mean axial speed leads to additional damping, especially in the higher order resonance. (3) The instability occurs more possibly in low-order principal parametric resonance, and the beam viscoelasticity effect more on the high-order principal parametric resonance. Acknowledgement This work was supported by the National Natural Science Foundation of China (Project Nos. 10672092 and 10725209), Shanghai Municipal Education Commission Scientific Research Project (No. 07ZZ07), Innovation Foundation for Graduates of Shanghai University (Project No. A.16-0101-07-011), and Shanghai Leading Academic Discipline Project (No. Y0103).

1120

H. Ding, L.-Q. Chen / European Journal of Mechanics A/Solids 27 (2008) 1108–1120

References Abrate, A.S., 1992. Vibration of belts and belt drives. Mechanism and Machine Theory 27, 645–659. Boresi, A.P., Chong, K.P., Saigal, S., 2003. Approximate Solution Methods in Engineering Mechanics. Wiley, New York. pp. 42–47. Chen, L.Q., Yang, X.D., 2005. Stability in parametric resonances of an axially moving viscoelastic beam with time-dependent velocity. Journal of Sound and Vibration 284, 879–891. 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. Chen, L.Q., Yang, X.D., Cheng, C.J., 2004. Dynamic stability of an axially accelerating viscoelastic beam. European Journal of Mechanics A Solids 23, 659–666. Chen, L.Q., Zu, J.W., 2008. Solvability condition in multi-scale analysis of gyroscopic continua. Journal of Sound and Vibration 309 (1–2), 338– 342. ˇ Cepon, G., Boltežar, M., 2007. Computing the dynamic response of an axially moving continuum. Journal of Sound and Vibration 300, 316–329. Marynowski, K., Kapitaniak, T., 2002. Kelvin–Voigt versus Burgers internal damping in modeling of axially moving viscoelastic web. International Journal of Non-Linear Mechanics 37, 1147–1161. Marynowski, K., 2004. Non-linear vibrations of an axially moving viscoelastic web with time-dependent tension. Chaos, Solitons and Fractals 21, 481–490. Marynowski, K., 2006. Two-dimensional rheological element in modeling of axially moving viscoelastic web. European Journal of Mechanics A Solids 25, 729–744. Marynowski, K., Kapitaniak, T., 2007. Zener internal damping in modeling of axially moving viscoelastic beam with time-dependent tension. International Journal of Non-Linear Mechanics 42, 118–131. Mockensturm, E.M., Guo, J., 2005. Nonlinear vibration of parametrically excited, viscoelastic, axially moving strings. Journal of Applied Mechanics 72, 374–380. Öz, H.R., 2001. On the vibrations of an axially traveling beam on fixed supports with variable velocity. Journal of Sound Vibration 239, 556–564. Öz, H.R., Pakdemirli, M., 1999. Vibrations of an axially moving beam with time dependent velocity. Journal of Sound Vibration 227 (2), 239–257. Öz, H.R., Pakdemirli, M., Özkaya, E., 1998. Transition behavior from string to beam for an axially accelerating material. Journal of Sound Vibration 215, 571–576. Özkaya, E., Öz, H.R., 2002. Determination of natural frequencies and stability regions of axially moving beams using artificial neural networks method. Journal of Sound Vibration 252 (4), 782–789. Özkaya, E., Pakdemirli, M., 2000. Vibrations of an axially accelerating beam with small flexural stiffness. Journal of Sound Vibration 234, 521–535. Parker, R.G., Lin, Y., 2001. Parametric instability of axially moving media subjected to multifrequency tension and speed fluctuations. ASME Journal of Applied Mechanics 68 (1), 49–57. Pasin, F., 1972. Über die Stabilität der Beigeschwingungen von in laengsrichtung periodisch hin und herbewegten Stäben. Ingenieur-Archiv 41, 387–393. Suweken, G., Van Horssen, W.T., 2003. On the transversal vibrations of a conveyor belt with a low and time-varying velocity, Part II: The beam like case. Journal of Sound and Vibration 267, 1007–1027. Wickert, J.A., Mote, C.D. Jr., 1988. Current research on the vibration and stability of axially moving materials. Shock and Vibration Digest 20 (5), 3–13. Yang, X.D., Chen, L.Q., 2005. Bifurcation and chaos of an axially accelerating viscoelastic beam. Chaos, Solitons and Fractals 23, 249–258. Yang, X.D., Chen, L.Q., 2006. Stability in parametric resonance of axially accelerating beams constituted by Boltzmann’s superposition principle. Journal of Sound and Vibration 289, 54–65.