12
Click here to load reader
Click here to load reader
Appl. Math. Mech. -Engl. Ed., 33(6), 817–828 (2012)DOI 10.1007/s10483-012-1588-8c©Shanghai University and Springer-Verlag
Berlin Heidelberg 2012
Applied Mathematicsand Mechanics(English Edition)
Asymptotic analysis on weakly forced vibration of axially movingviscoelastic beam constituted by standard linear solid model∗
Bo WANG (� �)
(School of Mechanical Engineering, Shanghai Institute of Technology,
Shanghai 200235, P. R. China)
Abstract The weakly forced vibration of an axially moving viscoelastic beam is inves-tigated. The viscoelastic material of the beam is constituted by the standard linear solidmodel with the material time derivative involved. The nonlinear equations governingthe transverse vibration are derived from the dynamical, constitutive, and geometricalrelations. The method of multiple scales is used to determine the steady-state response.The modulation equation is derived from the solvability condition of eliminating secularterms. Closed-form expressions of the amplitude and existence condition of nontrivialsteady-state response are derived from the modulation equation. The stability of non-trivial steady-state response is examined via the Routh-Hurwitz criterion.
Key words axially moving beam, weakly forced vibration, standard linear solid model,method of multiple scales, steady-state response
Chinese Library Classification O3262010 Mathematics Subject Classification 74G10, 74H10, 74K10
1 Introduction
As an important mechanical model, axially moving beams can represent many engineeringdevices, such as band saws, power transmission belts, aerial cable tramways, crane hoist ca-bles, flexible robotic manipulators, and spacecraft deploying appendages[1]. Despite usefulnessand advantages of these devices, the vibrations associated with the devices have limited theirapplications. Therefore, understanding the transverse vibrations of axially moving beams isimportant for the designs of the devices.
The transverse parametric vibrations of axially accelerating elastic beams have been exten-sively analyzed since the first study by Pasin[2]. Oz et al.[3] employed the method of multiplescales to study the dynamic stability of an axially accelerating beam with small bending stiff-ness. Oz[4] used the method of multiple scales to analytically calculate the stability boundariesof an axially accelerating beam under pinned-pinned and clamped-clamped conditions, respec-tively. Suweken and van Horssen[5] used the method of multiple scales to a discretized system
∗ Received May 9, 2011 / Revised Feb. 29, 2012Project supported by the National Natural Science Foundation of China (No. 10972143), the Shang-hai Municipal Education Commission (No. YYY11040), the Shanghai Leading Academic DisciplineProject (No. J51501), the Leading Academic Discipline Project of Shanghai Institute of Technology(No. 1020Q121001), and the Start Foundation for Introducing Talents of Shanghai Institute of Tech-nology (No. YJ2011-26)Corresponding author Bo WANG, Ph.D., E-mail: [email protected]
818 Bo WANG
via the Galerkin method to study the dynamic stability of an axially accelerating beam withpinned-pinned ends. Pakdemirli and Oz[6] employed the method of multiple scales to analyzethe stability in the resonances involved up to four modes.
In addition, for elastic beams, the axially accelerating viscoelastic beams have recently beeninvestigated. Chen et al.[7] used the averaging method to a discretized system via the Galerkinmethod to analytically present the stability boundaries of the axially accelerating viscoelasticbeams with clamped-clamped ends. Chen and Yang[8] used the method of multiple scaleswithout discretization to analytically obtain the stability boundaries of the axially acceleratingviscoelastic beams with pinned-pinned or clamped-clamped ends. Yang and Chen[9] used themethod of multiple scales to analytically present the vibration and stability of the axiallymoving beam constituted by the viscoelastic constitutive law of an integral type. Chen andYang[10] used the method of multiple scales to analytically present the vibration and stabilityof an axially moving beam constrained by the simple supports with rotational springs. In theworks of Chen et al.[7] and Chen and Yang[8,10], the Kelvin model containing the partial timederivative was used to describe the viscoelastic behavior of beam materials.
Geometrical nonlinearity caused by the finite stretching of beams cannot be neglected whenlarge transverse displacement occurs. Maccari[11] determined the external force-response andfrequency-response curves in the cases of primary resonance and subharmonic resonance for aweakly periodically forced beam with the quadratic and cubic nonlinearities. Boertjens andvan Horssen[12] studied the interactions of modes for a weakly forced beam with the geomet-ric nonlinearity. Chen et al.[13] analyzed the nonlinear vibration of a cantilever in a contactatomic force microscope with a nonlinear boundary via an asymptotic approach. However,those investigators focused on the vibration of stationary beams, which are disturbed by con-servative continuous systems. Chen and Yang[14] investigated the steady-state response andtheir stability of two nonlinear models of the axially moving viscoelastic beams. Yang et al.[15]
adopted the one-term Galerkin discretization and the perturbation method to study the forcedvibration of an axially accelerating beam subjected to the multi-frequency excitations. Dingand Chen[16] investigated the transverse forced vibration of axially moving viscoelastic beamsfor two nonlinear models via the finite difference method. Chen and Ding[17] used the finitedifference scheme to investigate steady-state periodical response for the planar vibration of theaxially moving viscoelastic beams subjected to external transverse loads.
Compared with the Kelvin model, the standard linear solid model is more typical andrepresentative. Meanwhile, this model can be degenerated to the Kevin or Maxwell model byvarying the alternative of the stiffness of beams. In addition, if the viscoelastic materials areconstituted by Botlzmann’s superposition principle with the relaxation modulus expressed bythe exponential function, the governing equation has the similar form to the standard linearsolid model[9]. Mockensturm and Guo[18] convincingly argued that the Kelvin model generalizedto the axially moving materials should contain the material time derivative to account forthe energy dissipation in the steady motion. Actually, the material time derivative was alsoemployed in the Kelvin model of axially moving materials by Chen and Yang[19]. In the presentpaper, the author adopts the standard linear solid model for two nonlinear models of the axiallyaccelerating viscoelastic beam by using the material time derivative to represent the beamviscoelastic material property. The method of multiple scales is used to obtain the solutions ofgoverning equations, the steady-state response, and their stability.
An outline of the paper is as follows. In the next section, the governing equations of motionbased on the standard linear solid model are derived from the Newton’s second law. Subsequentsection of the paper uses the method of multiple scales to obtain the approximate solution forthe primary resonance of the nth mode of the nonlinear system. In Section 4, the modulationequation of the amplitude and phase of the steady-state response is derived. Here, the stabilityof the frequency-response for the primary resonance is investigated.
Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam 819
2 Governing equations of motion
Consider a uniform axially moving viscoelastic beam with the density ρ, the cross-sectionalarea A, and the initial tension P0 travelling at an axial constant mean speed γ between twotransversely motionless ends separated by the distance l under the distributed external periodicexcitation F cos(ωt). Consider only the bending vibration of the beam in a reference framedescribed by the transverse displace v(x, t), where t is the time, and x is the axial coordinate.The physical model is shown in Fig. 1, and the equation of motion in the transverse directioncan be derived from the Newton’s second law as
ρAd2v
dt2+ M,xx =
∂
∂x((P0 + Aσ)v,x) + F cos(ωt), (1)
where (·),x denotes the partial differentiation with respect to x. The excitation amplitude Fand the excitation frequency ω are constant. σ(x, t) is the disturbed beam stress, and thebending moment M(x, t) is defined as
M(x, t) = −∫
A
zσ(x, z, t)dA, (2)
where the zx-plane is regarded as the principal plane of bending, and σ(x, z, t) is the normalstress.
Fig. 1 Physical model of axially moving beam
The standard linear solid model is adopted to describe the viscoelastic property of the beammaterial. The stress-strain relationship of the model is expressed in a differential form as[20–23]
(E1 + E2)σ(x, z, t) + ηddt
σ(x, z, t) = E1E2ε(x, z, t) + E1ηddt
ε(x, z, t), (3)
where both E1 and E2 are elastic moduli, ε(x, z, t) is the axial strain, and η is the viscousdamping of the dashpot. The standard linear solid model, shown in Fig. 2, can be employedto describe the behavior of linear viscoelastic materials of solid type with the limited creepdeformation. It can be reduced to the Kelvin model (when E1 → ∞ and E2 �= 0) or theMaxwell model (when E1 �= 0 and E2 = 0).
Fig. 2 Standard linear solid model
For small deflections, the displacement-strain relation can be written as
ε(x, z, t) = −z∂2v(x, t)
∂x2. (4)
820 Bo WANG
The Lagrangian strain is employed as a finite measure to account for the geometric non-linearity due to the small but finite stretching of beams. For one-dimensional problems, thedisturbed stress σ(x, t) in (1) is still described by the standard linear solid model as
(E1 + E2)σ(x, t) + ηddt
σ(x, t) = E1E2εL(x, t) + E1ηddt
εL(x, t), (5)
where εL(x, t) is the Lagrangian strain, i.e.,
εL =12v,2x . (6)
Introduce the material time derivative by defining the differential operator ddt as[23]
ddt
↔ ∂
∂t+ γ
∂
∂x, (7)
where the speed γ is equal to dxdt . In fact, (7) is also regarded as the total time derivative.
Inserting (7) in (1), (3), and (5) yields
ρA(v,tt +2γv,xt +γ2v,xx) + M,xx = ((P0 + Aσ)v,x),x +F cos(ωt), (8)
(E1 + E2)σ + η(σ,t +γσ,x) = E1(E2ε + η(ε,t +γε,x)), (9)
(E1 + E2)σ + η(σ,t +γσ,x) = E1(E2εL + η(εL,t +γεL,x)). (10)
Eliminating ε from (9) via (4) and εL from (10) via (6), respectively, can lead to
(E1 + E2)σ + η(σ,t +γσ,x) = −zE1(E2v,xx + η(v,xxt + γv,xxx)), (11)
(E1 + E2)σ + η(σ,t +γσ,x) =12E1(E2(v,2x) + η(v,2x),t +γη(v,2x),x). (12)
The dynamical relation (8) with the constitutive relation (11) and the geometric relation (12)governs the equation of motion of an axially moving viscoelastic beam. If the spatial variationof the tension is rather small compared with the initial tension, the exact form of the disturbedtension Aσ can be replaced by its spatially averaged value[14,23−25]
1l
∫ l
0
Aσdx.
Then, (8) can be replaced by
ρA(v,tt +2γv,xt +γ2v,xx) + M,xx =(P0 +
1l
∫ l
0
Aσdx)v,xx +F cos(ωt). (13)
In order to nondimensionalize and eliminate σ and z, multiplying both sides of (11) with zP0l
and integrating the resulting equation yield
1P0l
((E1 + E2)
∫A
zσdA + η(∫
A
zσ,tdA + γ
∫A
zσ,xdA))
= − E1I
P0l(E2v,xx + η(v,xxt + γv,xxx)), (14)
where I is the moment of inertial, which is expressed as follows:
I =∫
A
z2dA. (15)
Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam 821
Multiplying both sides of (12) with AP0
leads to
A
P0((E1 + E2)σ + η(σ,t +γσ,x)) =
12
AE1
P0(E2(v,2x) + η(v,2x),t +γη(v,2x),x). (16)
Let the dimensionless variables be⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
v ↔ v√εl
, x ↔ x
l, t ↔ t
√P0
ρAl2, c = γ
√ρA
P0, ω ↔ ω
√ρAl2
P0, f =
Fl
εP0,
ς(x, t) =1
P0l
∫A
zσ(x, z, t)dA, ζ(x, t) =Aσ
P0, α =
η
ε(E1 + E2)
√P0
ρAl2,
Ea =IE1E2
P0l2(E1 + E2), Eb =
IE1
P0l2, Ec =
E1E2A
P0(E1 + E2), Ed =
E1A
P0,
(17)
where the book-keeping device ε is a small dimensionless parameter accounting for the factthat the transverse displacement, the viscous damping, and the nonlinearity and excitationamplitude are very small. Here, we regard the action of the small excitation amplitude as theweak external excitation. Furthermore, F has the same order as ε. Equations (8), (14), and(16) can be cast into the dimensionless forms of the governing equations, respectively, i.e.,
v,tt +2cv,xt +(c2 − 1)v,xx −ς,xx = ε(ζv,x),x +εf cos(ωt), (18)
ς + εας,t +εαcς,x = −Eav,xx −εαEb(v,xxt +cv,xxx), (19)
ζ + εαζ,t +εαcζ,x =12Ecv,2x +
12εαEd((v,2x),t +c(v,2x),x). (20)
We suppose that the beam is constrained at both ends by simple supports, and the boundarycondition is expressed in the dimensionless form as follows:
v(0, t) = 0, v(1, t) = 0, v,xx (0, t) = 0, v,xx (1, t) = 0. (21)
3 Approximate solution for primary resonance of nth mode
In this section, we shall use the method of multiple scales to solve the governing equation(18) with (19) and (20) subjected to the boundary condition (21), defining a slow time scaleT = εt and looking for the asymptotic solutions of the form, respectively, i.e.,
v(x, t; ε) = v0(x, t, T ) + εv1(x, t, T ) + O(ε2), (22)
ς(x, t; ε) = ς0(x, t, T ) + ες1(x, t, T ) + O(ε2), (23)
ζ(x, t; ε) = ζ1(x, t, T ) + O(ε). (24)
Inserting (22) into (18) and equating the coefficients of each power of ε to be zero, we obtainthe equations below at the orders ε0 and ε1 as
v0,tt +2cv0,xt +(c2 − 1)v0,xx −ς0,xx = 0, (25)
v1,tt +2cv1,xt +(c2 − 1)v1,xx −ς1,xx
= f cos(ωt) − 2v0,tT +v0,x ζ1,x +2γv1,xt +v0,xx ζ1. (26)
In the same way, inserting (23) into (19), we obtain the equations below at the orders ε0 andε1 as
ς0(x, t, T ) = −Eav0,xx , (27)
ς1 + ας0,t +αγς0,x = −Eav1,xx −αEbv0,xxt −αγEbv0,xxx . (28)
822 Bo WANG
Inserting (24) into (20), we obtain the equation below at the order ε0 as
ζ1(x, t, T ) = Ec(v0,x)2. (29)
Inserting (27) into (25) and substituting (27)–(29) into (26) yield
Mv0,tt +Gv0,t +Kv0 = 0, (30)Mv1,tt +Gv1,t +Kv1
= − 2v0,tT −2cv0,xT +f cos(ωt) +32Ec(v0,x)2v0,xx
+ α(Ea − Eb)(v0,xxxxt +cv0,xxxxx), (31)
where the mass, the gyroscopic, and the linear stiffness operators are, respectively, defined asfollows:
M = I, G = 2c∂
∂x, K = (c2 − 1)
∂2
∂x2+ Ea
∂4
∂x4. (32)
We assume that the system is in the primary resonance with the external excitation. Inorder to express the nearness of the excitation frequency to the natural frequencies, we definea detuning parameter σ through the relation
ω = ωn + εσ, (33)
where ωn is the nth natural frequency of the linear undamped and unforcecd gyroscopic con-tinuous system (30) subjected to the boundary condition (21).
The assumed solution to (30) has been given as follows:
v0(x, t, T ) = ϕn(x)An(T )eiωnt + cc, (34)
where cc stands for the complex conjugate of all preceding terms on the right-hand side of(34), ϕn(x) is the modal function[10], An(T ) is an arbitrary function of time, and ϕn(x) can bedefined as
ϕn(x) = eir1nx − (r24n − r2
1n)(eir3n − eir1n)(r2
4n − r22n)(eir3n − eir2n)
eir2nx − (r24n − r2
1n)(eir2n − eir1n)(r2
4n − r23n)(eir2n − eir3n)
eir3nx
−(1 − (r2
4n − r21n)(eir3n − eir1n)
(r24n − r2
2n)(eir3n − eir2n)− (r2
4n − r21n)(eir2n − eir1n)
(r24n − r2
3n)(eir2n − eir3n)
)eir4nx, (35)
where rin (i =1, 2, 3, 4) are four roots of the following fourth-order algebraic equation:
Ear4in − (c2 − 1)r2
in − 2cωnrin − ω2n = 0. (36)
Substituting (33) and (34) into (31), we obtain
Mv1,tt +Gv1,t +Kv1
=(−2(iωnϕn + cϕ′
n)An + αAn(Ea − Eb)(iωnϕ(4)n + cϕ(5)
n )
+32EcA
2nAn(2ϕ′
nϕ′nϕ′′
n + ϕ′2n ϕ′′
n) +12eiσT f
)eiωnt + TNS + cc, (37)
where the dot and the prime denote the differentiation with respect to T and x, respectively,and TNS stands for the terms that will not bring secular terms into the solution. Equation(37) has a bounded solution only if the solvability condition holds. The solvability conditiondemands the following orthogonal relationships[26]:
⟨−2An(iωnϕn + cϕ′
n) +12eiσT f +
32EcA
2nAn(2ϕ′
nϕ′nϕ′′
n + ϕ′nϕ′′
n)
+ Anα(Ea − Eb)(iωnϕ(4)n + cϕ(5)
n ), ϕn
⟩= 0, (38)
Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam 823
where the inner product 〈g1, g2〉 for the complex functions g1 and g2 on [0, 1] is defined as
〈g1, g2〉 =∫ 1
0
g1(x)g2(x)dx. (39)
The application of the distributive law of the inner product to (38) leads to
An + μnα(Eb − Ea)An + κnEcAnA2n + χneiσT f = 0, (40)
where ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
μn =c
∫ 1
0
ϕnϕ(5)n dx + iωn
∫ 1
0
ϕnϕ(4)n dx
2∫ 1
0
(iωnϕn + cϕ′n)ϕndx
,
χn = −
∫ 1
0
ϕndx
4∫ 1
0
(iωnϕn + cϕ′n)ϕndx
,
κn =−3
2
∫ 1
0
ϕnϕ′2n ϕ′′
ndx − 3∫ 1
0
ϕnϕ′nϕ′
nϕ′′ndx
2∫ 1
0
(iωnϕn + cϕ′n)ϕndx
.
(41)
Apparently, (41) is determined by the natural frequencies and the modal function of the linearsystem (30) subjected to the boundary condition (21), which relates to the transport speed cand the stiffness constants Ea and is independent of the viscous damping α and the nonlinearityEc. It can be numerically verified that
Re(μn) > 0, Im(μn) = 0; Re(κn) = 0, Im(κn) < 0. (42)
Actually, the primary resonance in the system governed by (13) can be analyzed via the asymp-totic analysis in a similar way. The solvability condition is still expressed by (40) with thecoefficients given by (41), while κn is redefined by
κn =−1
2
∫ 1
0
ϕ′2n dx
∫ 1
0
ϕnϕ′′ndx −
∫ 1
0
ϕ′nϕ′
ndx
∫ 1
0
ϕnϕ′′ndx
2∫ 1
0
(iωnϕn + cϕ′n)ϕndx
. (43)
4 Steady-state response and stability with numerical examples
Expressing the complex-valued function An(T ) into real and imaginary parts as follows:
An(T ) = an(T )eiβn(T ), (44)
where αn(T ) and βn(T ) are, respectively, the real amplitude and phase of the response. Substi-tuting (44) into (40) and separating it into real and imaginary parts, we arrive at the followingmodel equations:
⎧⎨⎩
an = f(Im(χn) sin θn − Re(χn) cos θn) + αRe(μn)(Ea − Eb)an = Φ(an, θn),
θn =f
an(Im(χn) cos θn + Re(χn) sin θn) + σ + Im(κn)Eca
2n = Ψ(an, θn),
(45)
824 Bo WANG
where
θ = σT − βn. (46)
For the steady-state response, the amplitude αn and the new phase angle θn in (45) are constant.Setting αn=α0n and θn=θ0n, we obtain⎧⎨
⎩0 = f(Im(χn) sin θ0n − Re(χn) cos θ0n) + αRe(μn)(Ea − Eb)a0n,
0 = f(Im(χn) cos θ0n + Re(χn) sin θ0n) + σa0n + Im(κn)Eca30n.
(47)
Apparently, there exists no trivial solution from (47). Eliminating θ0n leads to
W (a0n, σ) = (α(Eb − Ea)Re(μn)a0n)2 + (Im(κn)Eca30n − σa0n)2 − f2|χn|2 = 0. (48)
Here, this expression is a modulation equation of amplitude and phase of the steady-stateresponse. Hence, the following two nontrivial solutions can be found:⎧⎪⎪⎨
⎪⎪⎩σ1 = −Im(κn)Eca
20n − 1
a0n
√f2|χn|2 − (αRe(μn)(Eb − Ea)a0n)2,
σ2 = −Im(κn)Eca20n +
1a0n
√f2|χn|2 − (αRe(μn)(Eb − Ea)a0n)2.
(49)
In the following, the transverse displacement of the center of the beam is used to representthe periodical steady-state responses of the beam motion. Consider an axially moving beamwith Ea = 0.64 and c = 2.0 subjected to the boundary condition (21). The first two naturalfrequencies of the linear system (30) are ω1 = 5.369 2 and ω2 = 30.12. Equation (45) givesμ1 = 110.73, χ1 = 0.018 2+0.0082 i, μ2 = 1006.48, and χ2 = 0.000 9+0.0014 i. For the non-linear partial-differential model (8), κ1 = –61.8985 i, and κ2 = –156.8370 i from (41). For thenonlinear integro-partial-differential model (13), κ1 = –40.9617 i, and κ2 = –94.4142 i from(43), accordingly. In all figures, E1 = E2 = 3, and Eb = Ea(E1 + E2)/E2=1.28 via (17). InFigs. 3–6, we show the frequency-response curves for the first and second modes in the primaryresonance from (49), where the detuning σ is the function of the response an.
In Fig. 3, we show the frequency-response curves for the first and second modes with thedifferent viscous damping in the primary resonance. It illustrates that the amplitude of thesteady-state response becomes small with the viscous damping increasing apparently nearbythe natural frequency ωn (σ = 0).
Fig. 3 Effect of viscous damping
Figure 4 reveals the effect of force on the steady-state response in the first two primaryresonances. However, it illustrates that if the force increases, the amplitude of the steady-stateresponse will change largely not only around the natural frequency ωn but also away from it.
Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam 825
Fig. 4 Effect of force
Figure 5 depicts the effect of the nonlinear coefficient Ec on the steady-state response in thefirst two primary resonances. With the increasing nonlinear coefficient, the amplitude of thesteady-state response will become small.
Fig. 5 Effect of nonlinear coefficients
Figure 6 shows the frequency-response curves for the first and second modes of two nonlin-ear models in the primary resonance. With some specified parameters, the amplitude of thesteady-state response of the integro-partial-differential model is larger than that of the partial-differential model. It is easily found that Fig. 6 is similar to Fig. 5. The difference of two modelsis due to the nonlinear terms in (8) and (13). Besides, the nonlinear coefficient Ec also existsin (8) and (13).
Fig. 6 Effect of nonlinear model
826 Bo WANG
In all the above figures, the primary resonance phenomena are shown that, if the excitationfrequency approaches to the natural frequency, namely, the primary resonance occurs, thetransverse amplitude becomes rather large. In fact, if there is no nonlinearity and viscousdamping, the transverse amplitude will be infinitely large. The system turns into the steady-state due to their existence.
We shall proceed studying the stability of the steady-state response. Constructing theJacobian matrix ⎡
⎢⎢⎣∂Φ∂an
∂Ψ∂θn
∂Φ∂an
∂Ψ∂θn
⎤⎥⎥⎦
an=a0n, θn=θ0n,
(50)
using (47) when necessary, and evaluating the eigenvalues of this matrix, we have
λ2 + s1λ + s2 = 0, (51)
where
s1 = 2Re(μn)α(Eb − Ea), (52)
s2 = Re(μn)2α2(Eb − Ea)2 + (σ + Im(κn)Eca20n)(σ + 3Im(κn)Eca
20n). (53)
It turns out numerically that s2 is always positive with σ1. Hence, the σ1 curve is always stableaccording to the Routh-Hurwitz criterion, but there possibly exists an unstable region withinthe σ2 curve. A numerical example is shown in Fig. 7: the solid line stands for the stable regionof the frequency-response curve, the dotted line stands for the unstable boundary (s2=0), andthe dashed line stands for the unstable region. The σ2 curve can become always stable viaincreasing the viscous damping and decreasing the force.
Fig. 7 Frequency-response curve and stability for case of primary resonance
Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam 827
5 Conclusions
The weakly forced vibration of an axially accelerating beam constituted by the standardlinear solid model is devoted by using the material time derivative. The method of multiplescales is used to determine the steady-state response. Several conclusions are obtained via thefrequency-response curves.
(i) Increasing the viscous damping and/or nonlinear coefficient will make the amplitude ofthe steady-state response become small. They can restrain the large amplitude occurring inthe primary resonance. This effect is evident in the resonance region in particular.
(ii) The amplitude of the steady-state response will change largely with the increase in theforce, not only around the natural frequency but also away from it.
(iii) The integro-partial-differential model has larger transverse displacement than the partial-differential model under the same conditions.
(iv) Increasing the viscous damping and decreasing the force can make the steady-stateresponse become stable if there exists an unstable region.
References
[1] Evans, T. Nonlinear Dynamics, INTECH, Croatia, 145–172 (2010)
[2] Pasin, F. Ueber die stabilitat der beigeschwingungen von in laengsrichtung periodisch hin undherbewegten staben. Ingenieur-Archiv, 41, 387–393 (1972)
[3] Oz, H. R., Pakdemirli, M., and Ozkaya, E. Transition behavior from string to beam for an axiallyaccelerating material. Journal of Sound and Vibration, 215, 571–576 (1998)
[4] Oz, H. R. On the vibrations of an axially traveling beam on fixed supports with variable velocity.Journal of Sound and Vibration, 239, 556–564 (2001)
[5] Suweken, G. and van Horssen, W. T. On the transversal vibrations of a conveyor belt with alow and time-varying velocity, part II: the beam like case. Journal of Sound and Vibration, 267,1007–1027 (2003)
[6] Pakdemirli, M. and Oz, H. R. Infinite mode analysis and truncation to resonant modes of axiallyaccelerated beam vibrations. Journal of Sound and Vibration, 311, 1052–1074 (2008)
[7] Chen, L. Q., Yang, X. D., and Cheng, C. J. Dynamic stability of an axially accelerating viscoelasticbeam. European Journal of Mechanics-A/Solids, 23, 659–666 (2004)
[8] Chen, L. Q. and Yang, X. D. Stability in parametric resonances of an axially moving viscoelasticbeam with time-dependent velocity. Journal of Sound and Vibration, 284, 879–891 (2005)
[9] Yang, X. D. and Chen, L. Q. Stability in parametric resonance of axially accelerating beamsconstituted by Boltzmann’s superposition principle. Journal of Sound and Vibration, 289, 54–65(2006)
[10] Chen, L. Q. and Yang, X. D. Vibration and stability of an axially moving viscoelastic beam withhybrid supports. European Journal of Mechanics-A/Solids, 25, 996–1008 (2006)
[11] Maccari, A. The asymptotic perturbation method for nonlinear continuous systems. NonlinearDynamics, 19, 1–18 (1999)
[12] Boertjens, G. J. and van Horssen, W. T. On interactions of oscillation modes for a weakly non-linear undamped elastic beam with an external force. Journal of Sound and Vibration, 235,201–217 (2000)
[13] Chen, L. Q., Lim, C. W., Hu, Q. Q., and Ding, H. Asymptotic analysis of a vibrating cantileverwith a nonlinear boundary. Science in China Series G—Physics Mechanics Astronomy, 52, 1414–1422 (2009)
[14] Chen, L. Q. and Yang, X. D. Steady-state response of axially moving viscoelastic beams with pul-sating speed: comparison of two nonlinear models. International Journal of Solids and Structures,42, 37–50 (2005)
[15] Yang, T. Z., Fang, B., Chen, Y., and Zhen, Y. X. Approximate solutions of axially movingviscoelastic beams subject to multi-frequency excitations. International Journal of Non-LinearMechanics, 44, 230–238 (2009)
828 Bo WANG
[16] Ding, H. and Chen, L. Q. Nonlinear models for transverse forced vibration of axially movingviscoelastic beams. Shock and Vibration, 18, 281–287 (2011)
[17] Chen, L. Q. and Ding, H. Steady-state transverse response in coupled planar vibration of axiallymoving viscoelastic beams. Journal of Vibration and Acoustics, 132(1), 011009 (2010)
[18] Mockensturm, E. M. and Guo, J. Nonlinear vibration of parametrically excited, viscoelastic,axially moving strings. Journal of Applied Mechanics, 72, 374–380 (2005)
[19] Chen, L. Q. and Yang, X. D. Bifurcation and chaos of an axially accelerating viscoelastic beam.Chaos, Solitons and Fractals, 23, 249–258 (2005)
[20] Hou, Z. and Zu, J. W. Nonlinear free oscillations of moving belts. Mechanism and Machine Theory,37, 925–940 (2002)
[21] Fung, R. F., Huang, J. S., Chen, Y. C., and Yao, C. M. Nonlinear dynamic analysis of theviscoelastic string with a harmonically varying transport speed. Computers and Structures, 66,777–784 (1998)
[22] Ha, J. L., Chang, J. R., and Fung, R. F. Nonlinear dynamic behavior of a moving viscoelasticstring undergoing three-dimensional vibration. Chaos, Solitons and Fractals, 33, 1117–1134 (2007)
[23] Chen, L. Q. and Chen, H. Asymptotic analysis on nonlinear vibration of axially acceleratingviscoelastic strings with the standard linear solid model. Journal of Engineering Mathematics,67, 205–218 (2010)
[24] Chen, L. Q. and Ding, H. Steady-state responses of axially accelerating viscoelastic beams: ap-proximate analysis and numerical confirmation. Science in China Series G—Physics MechanicsAstronomy, 51, 1701–1721 (2008)
[25] Ding, H. and Chen, L. Q. On two transverse nonlinear models of axially moving beams. Sciencein China Series E—Technological Sciences, 52, 743–751 (2009)
[26] Chen, L. Q. and Jean, W. Z. Solvability condition in multi-scale analysis of gyroscopic continua.Journal of Sound and Vibration, 309, 338–342 (2008)
