European Journal of Mechanics A/Solids

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1 European Journal of Mechanics A/Solids 8 (009) Contents lists available at ScienceDirect European Journal of Mechanics A/Solids wwwelseviercom/locate/ejmsol Stability of axially accelerating viscoelastic beams: asymptotic perturbation analysis and differential quadrature validation Li-Qun Chen ab BoWang b a Department of Mechanics Shanghai University Shanghai China b Shanghai Institute of Applied Mathematics and Mechanics Shanghai University Shanghai 0007 China article info abstract Article history: Received 3 December 007 Accepted 7 December 008 AvailableonlineDecember008 Keywords: Asymptotic perturbation Differential quadrature Stability Parametric resonance Axially accelerating beam Viscoelasticity An asymptotic perturbation method is proposed to investigate stability of an axially accelerating viscoelastic beam The material time derivative is used in the viscoelastic constitutive relation The axial speed is characterized as a simple harmonic variation about the constant mean speed The stability condition can be determined via the asymptotic perturbation method The differential quadrature scheme is developed to solve numerically the equation of axially accelerating viscoelastic beams with simple supports The stability boundaries are numerically located in the summation parametric resonance and the principal parametric resonance Numerical examples show the effects of the beam viscoelasticity and the mean axial speed The numerical calculations validate the analytical results in the principal parametric resonance 008 Elsevier Masson SAS All rights reserved Introduction Axially moving beams can represent many engineering devices such as power transmission belts band saws and aerial cable tramways (Wickert and Mote 988; Abrate 99; Zwiers 007) As parametric vibration excited by the variation of the beam tension or the beam axial speed large transverse motion of axially moving beams may occur under certain conditions Transverse parametric vibration of axially accelerating elastic beams has been extensively analyzed since first study by Pasin (97) Öz et al (998) employed the method of multiple scales to study dynamic stability of an axially accelerating beam with small bending stiffness Özkaya and Pakdemirli (000) combined the method of multiple scales and the method of matched asymptotic expansions to construct nonresonant boundary layer solutions for an axially accelerating beam with small bending stiffness Öz and Pakdemirli (999) and Öz (00) applied the method of multiple scales to calculate analytically the stability boundaries of an axially accelerating beam under pinned-pinned and clampedclamped conditions respectively Parker and Lin (00) adopted a -term Galerkin discretization and the perturbation method to study dynamic stability of an axially accelerating beam subjected to a tension fluctuation Özkaya and Öz (00) used an artificial neural network algorithm to determine stability boundary of an axially accelerating beam Suweken and Horssen (003) ap- * Corresponding author address: lqchen@staffshueducn (L-Q Chen) plied 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 Zwiers and Braun (006a 006b) applied the Floquet s theory to a discretized system via the finite differences to conduct a numerical stability analysis for gyroscopic continuous systems Pakdemirli and Öz (008) employed the method of multiple scales to analyze the stability in the resonances involved up to four modes In addition to elastic beams axially accelerating viscoelastic beams have recently been investigated Chen et al (004) 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 (005) 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 (006) 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 (006) applied the method of multiple scales to present analytically vibration and stability of an axially moving beam constrained by simple supports with rotational springs In Chen et al (004) Chen and Yang ( ) the Kelvin model containing the partial time derivative was used to describe the viscoelastic behavior of beam materials Mockensturm and Guo (005) 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 /$ see front matter 008 Elsevier Masson SAS All rights reserved doi:006/jeuromechsol00800

2 L-Q Chen B Wang / European Journal of Mechanics A/Solids 8 (009) was also employed in the Kelvin model of axially moving materials by Marynowski and Kapitaniak (00) Marrynowski (004) Yang and Chen (005) and Marrynowski (006) aswellasinthe three-parameter viscoelastic model by Marynowski and Kapitaniak (007) Ding and Chen (008) revisited the problem addressed by Chen and Yang (006) by using the material time derivative in the Kelvin model Besides they considered the modes not involved in summation resonance in the analysis via the method of multiple scales and demonstrated that they have no effects on the stability Addition to the method multiple scales the method of asymptotic perturbation is also an effective approach to treat nonlinear vibration (Awrejcewicz et al 998; Andrianov et al 003) The asymptotic approach has been applied to vibrating beams Boertjens and van Horssen (998) constructed analytically approximate solutions in the cases of internal resonances for a beam with quadratic and cubic nonlinearities Maccari (999) determined external force-response and frequency-response curves in the cases of primary resonance and subharmonic resonance for a weakly periodically forced beam with quadratic and cubic nonlinearities Boertjens and van Horssen (000a) demonstrated the existence and uniqueness of solutions and the asymptotic validity of approximation for a beam with a quadratic nonlinearity Boertjens and van Horssen (000b) studied the interactions of modes for a weakly forced beam with a geometric nonlinearity Andrianov and Danishevs kyy (00) developed the asymptotic approach to locate periodic response of a beam with a cubic nonlinearity However those investigators focused on vibration of stationary beams which are disturbed conservative continuous systems There is few literature that is specially related to asymptotic approaches to an axially moving beam which is a disturbed gyroscopic continuous system To address the lack of research in this aspect this investigation presents an asymptotic perturbation analysis for an axially accelerating viscoelastic string 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 to confirm the analytical results via the numerical solutions to the governing equations Ding and Chen (008) studied the stability in principal parametric resonance of an axially accelerating viscoelastic beam based on the finite difference solution In the present investigation the authors develop a differential quadrature scheme to solve the governing equation numerically and to check the analytical results in both summation and principal parametric resonances The present paper is organized as follows Section presents the mathematical model Section 3 proposes an asymptotic perturbation approach to investigate stability in the model presented in Section Section 4 develops a differential quadrature scheme to solve the mode in Section and to predicts the effects of some parameters on the stability boundaries in the summation parametric resonance Section 5 compares the analytical and numerical results Section 6 ends the paper with the concluding remarks Problem formulations A uniform axially moving viscoelastic beam with density ρ cross-sectional area A momentofinertiali and initial tension P 0 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 V (X T ) T is the time and X is the axial coordinate The viscoelastic material obeys the Kelvin model with stiffness constant E and viscosity coefficient η The physical system is shown in Fig Then the governing equation in the dimensionless form is (Ding and Chen 008) Fig An axially accelerating beam v tt +γ v xt + γ v x + ( γ ) v xx +v f v xxxx + εαv xxxxt +εαγ v xxxxx = 0 () v = V L x = X L t = T P 0 ρ AL γ = v ρ A P 0 v f = EI P 0 L εα = Iη L 3 () ρ AP 0 and bookkeeping device ε is a small dimensionless parameter accounting for the fact that the viscosity coefficient is very small Assume that the beam is with simple supports at both ends Then the boundary conditions in dimensionless form are v(0 t) = 0 v xx (0 t) = 0 v( t) = 0 v xx ( t) = 0 (3) In the present investigation the axial speed is supposed to be a small simple harmonic variation about the constant mean speed γ (t) = γ 0 + εγ sin ωt (4) γ 0 is the mean axial speed and εγ and ω are respectively the amplitude and the frequency of the axial speed variation all in the dimensionless form Substitution of Eq (4) into Eq () yield Mv tt +Gv t +Kv= ε [ γ sin ωtv xt γ 0 γ sin ωtv xx ωγ cosωtv x αv xxxxt αγ 0 v xxxxx εγ sin ωtv xx εαγ sin ωtv xxxxx ] (5) the mass gyroscopic and linear stiffness operators are respectively defined as M = I G = γ 0 x K = ( γ 0 ) x + v f 3 Asymptotic perturbation analysis on stability 4 x 4 (6) For given boundary condition (3) the natural frequencies of the linear system corresponding to Eq (5) Mv tt +Gv t +Kv= 0 (7) can be determined Denote the frequencies as ω i (i = ) If the axial speed variation frequency ω approaches the sum of any two natural frequencies of Eq (7) the summation parametric resonance may occur A detuning parameter σ is introduced to quantify the deviation of ω from ω m + ω n (m n) and ω is described by ω = ω m + ω n + εσ (8) If ε 0 as ε is rather small to investigate the summation parametric response it is usually assumed that the response are mainly influenced by two corresponding modes and thus the effects of other modes can be neglected Therefore the solution to Eq (7) may take the form v(x t) = ψ m (x τ ε)e iω mt + ψ n (x τ ε)e iω nt + cc (9)

3 788 L-Q Chen B Wang / European Journal of Mechanics A/Solids 8 (009) τ = εt is the fast time scale and cc denotes the complex conjugate of all preceding terms on the right hand of an equation Functions ψ m (x τ ε) and ψ n (x τ ε) can be expanded in power series of ε ψ k (x τ ε) = ψ 0k (x τ ) + εψ k (x τ ) + O ( ε ) (k = mn) (0) The chain rule of partial derivatives leads to [ ψk e ±iω ] kt = (±iω k ψ k + εψ k τ )e ±iω kt (k = mn) () t Inserting Eqs (9) () into Eqs (5) and (3) and equating the coefficients of e iω kt (k = mn) attheorderε 0 and ε in the resulting equation yield at order ε 0 ω k Mψ 0k + iω k Gψ 0k + K ψ 0k = 0 (k = mn) () ψ 0k (0 τ ) = 0 ψ 0k xx (0 τ ) = 0 ψ 0k ( τ ) = 0 ψ 0k xx ( τ ) = 0 (k = mn) (3) at order ε ωm Mψ m + iω m Gψ m + K ψ m = ( iω m ψ 0m t γ 0 ψ 0m xt ) + (ω n ω m )γ ψ 0n x +iγ 0 γ ψ 0n xx e iστ iηω m ψ 0m xxxx ηγ 0 ψ 0m xxxx (4) ωn Mψ n + iω n Gψ n + K ψ n = ( iω n ψ 0n t γ 0 ψ 0n xt ) + (ω m ω n )γ ψ 0m x +iγ 0 γ ψ 0m xx e iστ iηω n ψ 0n xxxx ηγ 0 ψ 0n xxxx (5) ψ k (0 τ ) = 0 ψ k xx (0 τ ) = 0 ψ k ( τ ) = 0 ψ k xx ( τ ) = 0 (k = mn) (6) Assume the solution to Eq () is in the following form ψ 0k (x τ ) = q k (τ )φ k (x) (k = mn) (7) Then ω k Mφ k + iω k Gφ k + K φ k = 0 (k = mn) (8) φ k (0) = 0 φ k (0) = 0 φ k() = 0 φ k () = 0 (k = mn) (9) Under boundary (3) () has the solution (Öz and Pakdemirli 999; Chen and Yang 006) φ k (x) = e iβ kx (β 4k β k )(eiβ 3k e iβ k) (β 4k β k )(eiβ 3k e iβ k) e iβ kx (β 4k β k )(eiβ k e iβ k) (β 4k e iβ 3kx β 3k )(eiβ k e iβ 3k) [ (β 4k β k )(eiβ 3k e iβ k) (β 4k β k )(eiβ 3k e iβ k) (β 4k β k )(eiβ k e iβ ] k) (β 4k β 3k )(eiβ k e iβ 3k) e iβ 4kx (k = mn) (0) β jk ( j = 3 4; k = mn) are four roots of the following 4-order algebraic equation ω k γ 0ω k β k + ( γ 0 ) β k + v f β4 k = 0 (k = mn) () Substitution of Eq (7) into Eqs (4) and (5) leads to ωm Mψ m + iω m Gψ m + K ψ m = ( iω m φ m γ 0 φ m ) q m + (ω n ω m )γ φ n + iγ 0γ φ n q m e iστ ( iηω m φ m + ηγ 0φ ) m qm () ωn Mψ n + iω n Gψ n + K ψ n = ( iω n φ n γ 0 φ n ) q n + (ω n ω m )γ φ m + iγ 0γ φ m q n e iστ ( iηω n φ n + ηγ 0φ ) n qn (3) Introduce an inner product f f = 0 f (x) f (x) dx (4) for complex functions f and f defined on [0 ] Under the boundary conditions of vanishing the function values and the second-order x-partial derivatives both M and K are symmetric in the sense Mf f = f Mf Kf f = f Kf (5) and G is skew symmetric in the sense Gf f = f Gf (6) For function φ k (x) satisfying Eq (4) the distribution law of the inner product and Eqs (5) (6) and (4) yield ω k Mψ k + iω k Gψ k + K ψ k φ k = ψ k ω k Mφ k + iω k Gφ k + K φ k = 0 (k = mn) (7) Taking both hands of Eqs () and (3) inner product with φ k (x) (k = mn) and using Eq (7) give q m + ηc mm q m + γ d mn q n e iστ = 0 (8) q n + ηc nn q n + γ d nm q m e iστ = 0 (9) c kk = iω k 0 φ(4) φ k k dx + γ 0 0 φ(5) φ k k dx (iω k 0 φ φ k k dx + γ 0 0 φ φ k k dx) (k = mn) (30) d mn = (ω n ω m ) 0 n m dx + iγ 0 φ 0 n m dx 4(iω m 0 φ m φ m dx + γ 0 0 φ φ m m dx) (3) d nm = (ω m ω n ) 0 m n dx + iγ φ φ 0 0 m n dx 4(iω n 0 φ n φ n dx + γ 0 0 φ φ n n dx) (3) Eqs (8) and (9) are the same as those derived from the solvability condition via the method of multiple scales (Chen and Yang 006) while the coefficients defined by Eqs (30) (3) and (3) are different because the material time derivative is used in the constitutive relation Based on Eqs (8) and (9) one can develop the analytical expression of the stability boundary in summation parametric resonance (m n) and the principal parametric resonance (m = n) The details have been presented by Chen and Yang (006) and Ding and Chen (008) 4 Differential quadrature investigation on stability The differential quadrature method will be employed to solve numerically equation v tt +γ v xt + γ v x + ( γ ) v xx v f v xxxx + αv xxxxt +αγ v xxxxx = 0 (33)

4 L-Q Chen B Wang / European Journal of Mechanics A/Solids 8 (009) Eq (33) is the same as Eq () with the exception that ε = here Other numerical methods such as the Galerkin finite-element method (Čepon and Boltežar 007) and the finite difference method (Ding and Chen 008) may also serve the propose Introduce N sampling points as x = 0 x = δ x i = [ ] (i )π cos N 3 (i = 3 4N ) x N = δ x N = (34) The quadrature rules for the derivatives of a function at the sampling points yield v x (x i t) = v xxxx (x i t) = v xxxxx (x i t) = A () v(x j t) v xx (x i t) = A (4) v(x j t) A () v(x j t) A (5) v(x j t) (35) the weighting coefficients are the expression (Bert and Malki 996; Shu 00) N k=k i (x i x k ) A () = (x i x j ) N k=k j (x j x k ) and the recurrence relationship [ A (r) = r A (r ) ii A () A(r ) ] x i x j (i j = N; j i) (36) (r = 3 4 5; i j = N; j i) (37) A (r) = A (r) ii ik (r = 3 4 5; i = N) (38) k= k i Substitution of Eq (35) into Eqs (33) and (3) leads to ( v i + γ () A + η A (4) ) v j [ + γ () A + ( γ ) A () + v f A(4) + ηγ A (5) ] v j = 0 (i = 3N ) (39) v = v N = 0 A () j v j = A () N j v j = 0 (40) v i (t) = v(x i t) (4) Introduce N N mass matrix M gyroscopic matrix G and stiffness matrix K as follows M = (4) G = G 3 G 3 G 3N G 3N G N G N G N N G N N (43) A () A () A () N A () N K 3 K 3 K 3N K 3N K = K N K N K N N K N N A () N A () N A () N N A () N N G = γ A () + η A (4) (44) K = γ A () + ( γ ) A () + v f A(4) + ηγ A (5) (i = 3N ; j = N) (45) Then Eqs (39) and (40) can be cast into a discrete gyroscopic system M v + G v + Kv= 0 (46) v = (v v v N ) T (47) For the given parameters and initial conditions Eq (46) can be numerically solved via standard computation methods in structural dynamics After a time interval [0 T ] to remove the transient response the maximum beam center displacements V and V are respectively recorded for time intervals [T T ] and [T T ] If V is bigger than V the parametric resonance is stable If V is smaller than V the parametric resonance is unstable Varying the parameters one can locate the stability boundary in the parameter space In the present investigation the Newmark β-method (Newmark 959) is used to solve Eq(46) numerically Choose β = / The initial conditions for Eq (33) are chosen as v(x 0) = 0000x( x) v t (x 0) = 0 (48) In the differential quadrature method let N = 3 and δ = 0 5 To decide the stability choose T = 30 and T = 30 Consider an axially moving beam with v f = 08 η = 0000 and γ 0 = 0 Then ω = 5369 and ω = 3000 Fig shows the stability boundaries in the first principal parametric resonance (dashed line) the second principal parametric resonance (solid line) and the summation parametric resonance (dot line) In the first principal parametric resonance σ = ω ω In the second principal parametric resonance σ = ω ω In the summation parametric resonance σ = ω + ω ω The differential quadrature method can be employed to investigate the effects of the beam viscoelasticity and the mean axial speed on the stability boundary in the summation parametric resonance Fig 3 illustrates the effect of the beam viscoelasticity for γ 0 = 0 and η = 0000 (solid line) 0000 (dash-dotted line) (dot line) Fig 4 depicts the effect of the mean axial speed for η = 0000 and γ 0 = 0 (solid line) 5 (dash-dotted line) 0 (dot line) The larger viscosity coefficient and the smaller mean axial speed lead to the larger instability threshold of γ for given σ and the smaller instability range of σ for given γ That is the increasing viscosity coefficient and the decreasing mean axial

5 790 L-Q Chen B Wang / European Journal of Mechanics A/Solids 8 (009) Fig Stability boundaries in the first principal parametric resonance (dashed line) the second principal parametric resonance (solid line) and the summation parametric resonance (dot line) Fig 4 The effect of the mean axial speed on stability boundaries in summation parametric resonance Fig 3 The effect of the beam viscoelasticity on stability boundaries in summation parametric resonance speed make the stability boundaries move towards the increasing direction of γ in plane (ω γ ) and the instability regions become narrow The principal parametric resonance has the same changing tendencies of the stability boundary 5 Comparisons of analytical and numerical results Based on Eqs (8) and (9) derived from the asymptotic perturbation method the stability boundary in plane (σ γ )isdefined by the equation (Chen and Yang 006; Ding and Chen 008) [ + (c mm c nn ) ] σ + 4η (c mm + c nn ) c mm c nn + γ d nm d mn = 0 (49) Consider an axially moving beam with v f = 08 and γ 0 = 0 For ω = 5369 and ω = 3000 Eq () can be numerically solved the roots as β = β = i β 3 = i β 4 = 5058 and ω = β = β = i β 3 = i β 4 = 4650 Thus Eqs (30) (3) gives c = 073 c = d = i d = i d = i d = i Then Eq (49) leads to the stability boundaries obtained via the asymptotic perturbation in the first Fig 5 The comparison for the first principal parametric resonance principal parametric resonance (m = n = ) the second principal parametric resonance (m = n = ) and the summation parametric resonance (m = and n = ) In Fig 5 the stability boundaries via the asymptotic perturbation and the differential quadrature are compares in the first principal parametric resonance In the calculation η = 000 The comparison for the second principal parametric resonance is depicted in Fig 6 with η = 0000 The comparison for the summation parametric resonance is shown in Fig 7 with η = 0000 In all these figures the solid and dot lines represent the results of the asymptotic perturbation and the differential quadrature respectively Both Figs 5 and 6 demonstrate the differences in the first two principal parametric resonances are very small However Fig 7 indicates the difference in the summation parametric resonance is rather large 6 Conclusions This paper is devoted to parametric vibration of an axially accelerating beam constituted by the Kelvin model using the material time derivative The beam moves at an axial speed fluctuating harmonically about a constant mean speed An asymptotic perturbation method is proposed to determine the stability condition which is the same as that derived from the method of multiple

6 L-Q Chen B Wang / European Journal of Mechanics A/Solids 8 (009) Fig 6 The comparison for the second principal parametric resonance Fig 7 The comparison for the summation parametric resonance scales The differential quadrature scheme is developed to locate the stability boundary numerically The analytical results are validated by the numerical calculations in the principal parametric resonance Acknowledgements This work was supported by the National Outstanding Young Scientists Fund of China (Project No 07509) the National Natural Science Foundation of China (Project No 06509) Shanghai Municipal Education Commission Scientific Research Project (No 07ZZ07) and Shanghai Leading Academic Discipline Project (No Y003) References Abrate AS 99 Vibration of belt drives Mechanism and Machine Theory Andrianov IV Awrejcewicz J Barantsev RG 003 Asymptotic approaches in mechanics: new parameters and procedures Applied Mechanics Reviews Andrianov IV Danishevs kyy VV 00 Asymptotic approach for 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axially accelerating viscoelastic beam Chaos Solitons and Fractals Yang XD Chen LQ 006 Stability in parametric resonance of axially accelerating beams constituted by Boltzmann s superposition principle Journal of Sound and Vibration Zwiers U 007 On the Dynamics of Axially Moving Strings Cuvillier Zwiers U Braun M 006a Vibration analysis of gyroscopic systems Proceedings of Applied Mathematics and Mechanics Zwiers U Braun M 006b On the modeling of axially moving strings EUROMECH Newsletter