nonlinear dynamics of axially moving beam with coupled longitudinal–transversal vibrations

Nonlinear Dyn (2014) 78:2547–2556DOI 10.1007/s11071-014-1609-5

ORIGINAL PAPER

Nonlinear dynamics of axially moving beam with coupledlongitudinal–transversal vibrations

Xiao-Dong Yang · Wei Zhang

Received: 23 February 2014 / Accepted: 15 July 2014 / Published online: 2 August 2014© Springer Science+Business Media Dordrecht 2014

Abstract In this study, the nonlinear vibrations of anaxially moving beam are investigated by consideringthe coupling of the longitudinal and transversal motion.The Galerkin method is used to truncate the govern-ing partial differential equations into a set of couplednonlinear ordinary differential equations. By detuningthe axially velocity, the exact parameters with whichthe system may turn to internal resonance are detected.The method of multiple scales is applied to the govern-ing equations to study the nonlinear dynamics of thesteady-state response caused by the internal–externalresonance. The saturation and jump phenomena of suchsystem have been reported by investigating the nonlin-ear amplitude–response curves with respect to exter-nal excitation, internal, and external detuning parame-ters. The longitudinal external excitation may triggeronly longitudinal response when excitation amplitudeis weak. However, beyond the critical excitation ampli-tude, the response energy will be transferred from thelongitudinal motion to the transversal motion even theexcitation is employed on the longitudinal direction.Such energy transfer due to saturation has the potentialto be used in the vibration suppression.

X.-D. Yang (B) · W. ZhangCollege of Mechanical Engineering, Beijing Universityof Technology, Beijing 100124, Chinae-mail:[email protected]

Keywords Nonlinear vibrations · Axially movingbeam · Coupled longitudinal–transversal vibrations ·Multiple-scale method · Saturation phenomenon

1 Introduction

The axially moving structures are involved in manyengineering devices, such as belt drives, high-speedmagnetic tapes, band saws, fiber winding, and papersheets. The axial traveling speed plays an importantrole on the transverse vibrations of the structure andtheir dynamical stability. Study of the transverse vibra-tions becomes key to avoid possible resulting fatigue,failure, and low quality. String model and beam modelare usually used to investigate such structures. Recentdevelopments on axially moving continua have beenreviewed in [1–3].

The linear and nonlinear transverse dynamics of sys-tems with axially moving speed have been investigatedextensively in the literature. Wickert [4] examined thesub- and super-critical nonlinear vibrations of axiallymoving tensioned beams employing the linear integro-partial differential equation of motion. Marynowskiet al. [5,6] considered several energy dissipation mech-anisms in the mathematical model of axially movingsystems and investigated the bifurcation and chaos dueto the parametric excitation. The stability and vibra-tion characteristics of an axially moving plate wereinvestigated by Lin [7] by linear model. The dynam-ics of an axially moving beam beyond the first bifur-

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2548 X.-D. Yang, W. Zhang

cation were examined by Pellicano and Vestroni [8].The sub-critical vibrations of an axially moving beamwere investigated by Sze et al. [9] and Huang et al.[10] using the incremental harmonic balance method.Özhan and Pakdemirli [11] examined the primary res-onance of an axially moving beam by developing ageneral solution procedure for this class of systems.In a series of papers by Ghayesh et al. [12–20], thesub- and super-critical dynamics and stability of anaxially moving beam were investigated analytically andnumerically. Recently, nonlinear natural frequencies ofa high-speed moving beam were obtained by Ding andChen [21] using the Galerkin method.

It can be concluded from the above references thateven for the lower speed case the internal resonancemay induce complex dynamics for the axially movingbeam in sub-critical regime. Huang et al. [10] stud-ied the 3:1 internal resonance for the axially movingbeam. Özhan and Pakdemirli [11] discussed the non-linear dynamics of systems with cubic nonlinearitiesin the internal resonance case. Riedel and Tan [22]examined the coupled forced dynamics of an axiallymoving beam possessing a three-to-one internal reso-nance. Tang and Chen [23] investigated the nonlinearresponse of in-plane moving plate with the case of 3:1and 1:1 internal resonances. Ghayesh [24] studied thecomplex nonlinear dynamics, such as bifurcations andchaos, of axially moving beam for the 3:1 internal res-onance case. Wang et al. [25] considered the nonlineardynamics of the moving laminated circular cylindricalshells. The one-to-one internal resonance phenomenonwas discussed by the method of harmonic balance.

Most of the references studying internal resonancesconsidered only the transversal vibrations of the axi-ally moving material. Based on the Galerkin trunca-tion, the first two or more orders have been retained.When the second transversal mode natural frequencyapproaches the three times of the first transversal modenatural frequency by detuning the axial speed, the3:1 internal resonance phenomenon could be detectedfor the system with cubic nonlinearities. The coupledlongitudinal–transversal vibrations of axially movingsystems with constant axial speed, on the other hand,have not received considerable attention in the litera-ture. Ghayesh et al. [26–28] studied the longitudinal–transversal vibrations for the axially moving beam.They discussed the contribution of external excitationsto the nonlinear responses and also the 3:1 internal res-onances. However, the 2:1 internal resonances have not

been studied in the literature, which may yield interest-ing nonlinear dynamical phenomena.

In this study, the partial differential equations gov-erning the coupled longitudinal–transversal vibrationsare studied. The Galerkin method is used to truncatethe nonlinear partial differential equations into a set ofordinary differential equations. By detuning the axiallymoving speed, the internal resonance relations can befound. An example is proposed to illustrate the exis-tence of jump associated with saturation in the currentstudy for the axially moving beam.

2 Governing equations

Now, we consider the nonlinear dynamics of thelongitudinal–transversal coupling governing equationsfor the axially moving uniform beam with speed v ontwo simple supports with distance l. Assuming thatplane sections remain plane and assuming a linearstress–strain law, one can derive the following equa-tions of planar motion for beams [29]

ρ A

(∂2u

∂t2 + 2v∂2u

∂x∂t+ v2 ∂2u

∂x2

)+ c1

∂u

∂t− E A

∂2u

∂x2

= (E A − P)∂w

∂x

∂2w

∂x2 + F1 cos (�1t) (1)

ρ A

(∂2w

∂t2 + 2v∂2w

∂x∂t+ v2 ∂2w

∂x2

)

+ c2∂w

∂t− P

∂2w

∂x2 + E I∂4w

∂x4

= (E A − P)

(∂2u

∂x2

∂w

∂x+ ∂u

∂x

∂2w

∂x2

)+F2 cos (�2t)

(2)

where u is the longitudinal displacement; w is thetransversal displacement; ρ is the beam density; A andI are, respectively, the area and moment of inertia ofthe beam cross-section; E is Young’s modulus; P is theprescribed axial load; c1 and c2 are the constants relatedto the structural damping; and F1 and F2 are externalexcitation amplitudes. Equations (1) and (2) withoutthe traveling effect have been discussed by Nayfeh andMook [29] in detail. In the current version, the higherterms more than two have been omitted.

Equations (1) and (2) are coupled via the quadraticnonlinear terms. The similar equations governing themotions of beam without traveling have been investi-gated in Nayfeh and Mook [29] for the case of u =

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Nonlinear dynamics of axially moving beam 2549

O(w2) and the case of u = O(w). If the radius ofgyration of the cross-section be extremely small, thelongitudinal inertia is small compared with the restor-ing force, and it follows that u = O(w2); if the radiusof gyration be small, but not extremely small, it fol-lows that u = O(w). Ghayesh et al. [26–28] studiedthe coupled vibrations of axially moving material forthe case of u = O(w2). In the next section, it can beverified that by detuning the axially moving velocitywe can locate the vicinity where the transversal andlongitudinal motions have the same order, u = O(w),even the beam is a slender one.

3 Galerkin truncation

Modal analysis is one of the powerful tools to inves-tigate the vibration engineering problems. Accordingto the Galerkin truncation method, the time–spatialresponse function of the system can be expanded interms of linear undamped natural modes of the cor-responding time-independent system by neglecting theeffects of the damping, the axially moving velocity, andthe excitation. The response of our continuous systemin terms of the linear free-vibration modes for the cur-rent pinned–pinned ends conditions can be assumed asfollows:

u(x, t) =∞∑

n=1

pn(t) sinnπx

l(3)

w(x, t) =∞∑

n=1

qn(t) sinnπx

l(4)

Substituting (3) and (4) into (1) and (2) and mul-tiplying both sides of the results by sin (mπx/ l) andintegrate over the interval [0, l] lead to

pn + λ2n pn

= ε

{−μ1 pn − 4v

∞∑m=1

nm

m2 − n2

[(−1)m+n − 1

]pm

− κ

[ ∞∑m=1

mn (m + n) qmqm+n

+n∑

m=1

m2 (n − m) qmqn−m

]− (−1)n

nf1 cos (�1t)

}

(5)

qn + ω2nqn

= ε

{−μ2qn − 4v

∞∑m=1

nm

m2 − n2

[(−1)m+n − 1

]qm

− κ

[ ∞∑m=1

mn (m + n) (qm pm+n + pmqn+m)

+n∑

m=1

m2 (n − m) (pmqn−m + qm pn−m)

]

− (−1)n

nε f2 cos (�2t)

}(6)

where

μ1 = c1

ρ Al, μ2 = c2

ρ Al, κ = π3 E A − P

2ρ Al5,

f1 = 4F1l

ρ Aπ, f2 = 4F2l

ρ Aπ(7)

In this study, the amplitudes of excitation andresponses are considered small but finite, the structuredamping weak. The longitudinal and transversal nat-ural frequencies λn and ωn can be obtained by solvingfor the roots of the following two algebraic equation,respectively,

det(−λ2I + K1 + iλG

)= 0

det(−ω2I + K2 + iωG

)= 0 (8)

In Eq. (8), I is the unit matrix, G is the gyroscopicmatrix, and K1andK2 are diagonal stiffness matrices.The elements of G, K1, and K2 can be obtained as

(G)nm = 4vnm

m2 − n2

[(−1)m+n − 1

],

(K1)nn = n2π2

l2

(E

ρ− v2

),

(K2)nn = n4π4

l4

E I

ρ A− n2π2

l2 v2 + P

ρ A(9)

By studying the roots of (8), we discuss the longi-tudinal and transversal natural frequencies effected bythe axially moving velocity. Based on the parametersgiven by Table 1, the variation of the first two longitu-dinal and transversal natural frequencies with respectto the axial velocity is presented in Fig. 1. The naturalfrequencies decrease with the increase of the axiallymoving velocity, which has been discussed in manyreferences in the vibration study of axially movingmaterial [30–32]. The natural frequencies of the lon-gitudinal vibration are usually higher than those of thetransversal vibration when axially moving velocity islow. Although all the natural frequencies decrease withthe increase of the axially moving velocity, the values

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Table 1 The values of theparameters

E ρ A I l P

6.5 MPa 0.8 × 103kg/m3 1.0 × 10−3 m2 8.0 × 10−6 m4 1 m 500 N

Fig. 1 The natural frequencies versus the axially moving veloc-ity

for longitudinal and transversal natural frequencies arevarying with different rates.

By detuning the axially moving velocity, we canlocate some internal resonance points. For example,when velocity v = 9.46, the first longitudinal naturalfrequency λ1 = 355.78, and the first two transversal

natural frequencies are ω1 = 70.77 and ω2 = 28.501.In such case, the internal resonance may occur sinceλ1 = ω1 + ω2.

4 Analysis by the method of multiple scales

Now, we use the method of multiple scales to study thegoverning ordinary differential Eqs. (5) and (6) coupledvia the nonlinear terms. The approximate solutions forsmall but finite amplitudes can be assumed in the formof

pn (t) = pn0 (T0, T1) + εpn1 (T0, T1) + O(ε2

)

qn (t) = qn0 (T0, T1) + εqn1 (T0, T1) + O(ε2

)(10)

where T0 = t and T1 = εt represent the fast and slowtimescales. Substituting (10) and their derivatives into(5) and (6) and equating coefficients of like powers ofε, we obtain the following set of second-order ordinarydifferential equations:

O(ε0

):{

D20 pn0 + λ2

n pn0 = 0D2

0qn0 + ω2nqn0 = 0

(11)

O(ε1

):

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

D20 pn1 + ω2

n pn1 = −μ1D0 pn0 − 4v∞∑

m=1

nmm2−n2

[(−1)m+n − 1

]D0 pm0

− 2D0D1 pn0 − κ

[ ∞∑m=1

mn (m + n) qm0q(m+n)0 +n∑

m=1m2 (n − m) qm0q(n−m)0

]

− (−1)n

n f1 cos (�1t)

D20qn1 + γ 2

n qn1 = −μ2D0qn0 − 4v∞∑

m=1

nmm2−n2

[(−1)m+n − 1

]D0qm0

− 2D0D1qn0 − κ

[ ∞∑m=1

mn (m + n)(qm0 p(m+n)0 + pm0q(n+m)0

)

+n∑

m=1m2 (n − m)

(pm0q(n−m)0 + qm0 p(n−m)0

)] − (−1)n

n ε f2 cos (�2t)

(12)

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In the above equations, the differential operator Dn

denotes the derivative with respect to Tn .The solutions to (11) can be written as

pn0 = An (T1) exp (iλnT0) + An (T1) exp (−iλnT0)

qn0 = Bn (T1) exp (iωnT0) + Bn (T1) exp (−iωnT0)

(13)

Substituting (13) into (12) leads to

D20 pn1 + λ2

n pn1 = −μ1iλn Aneiλn T0

− 4v

∞∑m=1

nm

m2 − n2

[(−1)m+n − 1

]iλm Ameiλm T0

− 2D1 Aniλneiλn T0 − κ

∞∑m=1

mn (n + m)

{Bm Bn+mei(ωm+ωn+m )T0 + Bm Bn+mei(ωn+m−ωm )T0

}

− κ

n∑m=1

m2 (n − m){

Bm Bn−mei(ωm+ωn−m )T0

+Bm Bn−mei(ωm−ωn−m )T0}

− (−1)n

2nf1 exp (i�1T0) + cc (14)

D20qn1 + ω2

nqn1 = −μ2iωn Bneiωn T0

− 4v

∞∑m=1

nm

m2 − n2

[(−1)m+n − 1

]iωm Bmeiωm T0

− 2iωnD1 Bneiωn T0 − κ

∞∑m=1

mn (n + m)

[An+m Bmei(ωm+λn+m )T0 + An+m Bmei(λn+m−ωm )T0

]

− κ

∞∑m=1

mn (n + m)[

Am Bn+mei(λm+ωn+m )T0

+ Am Bn+mei(λm−ωn+m )T0]

− κ

n∑m=1

m2 (n − m)[

Am Bn−mei(λm+ωn−m )T0

+ Am Bn−mei(λm−ωn−m )T0]

− κ

n∑m=1

m2 (n − m)[

An−m Bmei(ωm+λn−m )T0

+ An−m Bmei(ωm−λn−m )T0]

− (−1)n

nf2 exp (i�2T0) + cc (15)

The cc symbol denotes the complex conjugate of allthe terms of the right-hand side. By considering thepossible secular terms of (14), we conclude that whenλn ≈ |ωn+m ± ωm | or λn ≈ |ωm ± ωn−m | extra inter-nal resonance link connecting pn1 and qn1 may exist.The similar conclusions can be obtained by studyingthe possible secular term of (15).

The nonresonant case and the internal resonant caseof the particular solutions of (14) and (15) need to bedistinguished, which will be discussed in the followingsubsection.

4.1 The nonresonant condition

In this case, the only terms that produce secular termsare the terms proportional to exp (±iλnT0) in (14) andthe terms proportional to exp (±iωnT0) in (15). Hence,the solvability conditions become

D1 An + 12μ1 An = 0

D1 Bn + 12μ2 Bn = 0 (16)

Substituting the solutions of (16) into (13)

pn0 = exp (−εμ1t)[an exp (iλnt) + cc

]qn0 = exp (−εμ2t)

[bn exp (iλnt) + cc

](17)

where an and bn are complex constants. There are onlyzero value steady-state solutions for (17), which is thetypical result of the damped free vibrations.

4.2 The resonant condition

In this subsection, we consider the coupled internal–external resonance conditions for the Eqs. (14) and (15)governing the longitudinal and transversal vibrationsof the beam due to the longitudinal excitation. Internalresonance may occur whenλn ≈ |ωn+m ± ωm |orλn ≈|ωm ± ωn−m |, and primary external resonance could bedescribed by �1 = λn . Now, we consider the internalresonance when λn ≈ ωn+m + ωm as a case study. Weintroduce the external and internal detuning parametersσ1 and σ2

�1 = λn + εσ1, λn = ωn+m + ωn + εσ2 (18)

To study the external–internal resonance better, weuse the case of n = 1, m = 1, f2 = 0 as an example.

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The first natural frequency of the longitudinal vibra-tion is detuned approximately equal to the sum of thefirst two natural frequencies of the transversal vibra-tion. The external excitation on the longitudinal direc-tion is focused while the excitation on the transversaldirection is omitted. In such case, the solvability con-ditions of (14) and (15) are

−iλ1 (2D1 A1 + μ1 A1) − 2κ B1 B2e−iσ2T1

+ 12 f1eiσ1T1 = 0

−iω1 (2D1 B1 + μ2 B1) − 2κ A1 B2eiσ2T1 = 0

−iω2 (2D1 B2 + μ2 B2) − 2κ A1 B1eiσ2T1 = 0. (19)

We introduce the polar form for the amplitudes as

An = 12 aneiαn , Bn = 1

2 bneiβn (20)

where an, bn, αn , and βn are real functions of T1.Substituting (20) into (19), separating real and imag-

inary parts of the resulting equations, and doing somemanipulations, we obtain the following differentialequations:

a1 = −1

2μ1a1 + κ

λ1b1b2 sin γ2 + f1

4λ1sin γ1

b1 = −1

2μ2b1 − κ

ω1a1b2 sin γ2

b2 = −1

2μ2b2 − κ

ω2a1b1 sin γ2

γ1 = σ1 − κ

λ1

b1b2

a1cos γ2 + f1

4λ1a1cos γ1, (21)

γ2 = σ2 + κ

λ1

b1b2

a1cos γ2 − f1

4λ1a1cos γ1

− κ

ω1

a1b2

b1cos γ2 − κ

ω2

a1b1

b2cos γ2

where the dot over any variables denotes the derivativewith respect to T1. In Eq. (21), two new variables areintroduced

γ1 = σ1T1 − α1, γ2 = σ2T1 + α1 − β1 − β2. (22)

To study the steady-state response, we set time deriv-atives to be zero and time-dependent variables to beconstants in (21) and (22). The steady-state solutionscan be obtained from the algebraic equations by lettingthe right-hand side to zero. By the set of algebraic equa-tions, the amplitude response curve can be obtained. Itcan be found that there are two possible steady-statesolutions: either a1 �= 0 and b1 = b2 = 0, or a1, b1

and b2 are all nonzero. For the first case b1 = b2 = 0,the analytical solution of a1 can be obtained:

a1 = f1

2λ1(μ2

1 + 4σ 21

)1/2 . (23)

For the case of a1, b1, and b2 all nonzero, the solutionsare

a1 =[(σ1 + σ2)

2 + 14 (μ2 + μ2)

2]1/2

(2√

κ2

ω1ω2

)

= 1

2κ

{ω1ω2

[(σ1 + σ2)

2 + 14 (μ1 + μ2)

2]}1/2

(24)

b1 =[χ (ω2/ω1)

1/2]1/2

, b2 =[χ (ω1/ω2)

1/2]1/2

.

(25)

In (25), the values of χ are determined by the roots ofthe following quadratic equation

κ2

λ21

χ2 + a1κ

λ1(μ1 sin γ2 − 2σ1 cos γ2) χ

+ 1

4μ2

1a21 + a2

1σ 21 −

(f1

4λ1

)2

= 0 (26)

where

sin γ2 = −μ2

/[(σ1 + σ2)

2 + μ22

]1/2

cos γ2 = (σ1 + σ2)

/[(σ1 + σ2)

2 + μ22

]1/2. (27)

The stability of steady-state solutions (23), (24), and(25) can be determined by the eigenvalues of the lin-earized coefficients matrix of the system (21) near thecorresponding steady-state solution. If the real part ofeach eigenvalue of the coefficient matrix is not pos-itive, then the corresponding steady-state solution isstable otherwise is unstable. In the following section,the stability of the steady-state solutions represented byamplitude responses will be checked by such criteria.

5 Numerical examples

Based on steady-state solutions (23)–(25), some numer-ical figures are presented to show the nonlinear phe-nomenon found in such system.

In Fig. 2, the first-order longitudinal amplitude a1

and the first- and second-order transversal amplitudes

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Fig. 2 Amplitudes of response as functions of the amplitude ofthe excitation; σ1 = 0.005; σ2 = 0

b1 and b2 are plotted as functions of the longitudi-nal excitation amplitude f1, where stable solutionsare indicated in solid lines while unstable solutions indashed lines. The internal resonance is perfectly tunedσ2 = 0, and the external resonance is slightly detunedas σ1 = 0.005.

In Fig. 2, one can clearly find the phenomenon ofsaturation. As longitudinal external excitation ampli-tude f1 increases from zero, so does the first-orderlongitudinal amplitude response a1 indicated in blacksolid line. This natural relation is determined by thelinear equation of (23). This agrees with the solutionof the corresponding linear problem. Beyond a criticalvalue, the solution of (23) loses stability (black dashedline) and another branch of solution determined by (24)dominates. It is clear that the first-order longitudinalresponse a1 of (24) is independent of external longitu-dinal excitation amplitude f1. Hence, the longitudinalresponse a1 holds the maximum value even for furtherincreases in f1. The longitudinal vibration mode is sat-urated.

Beyond the saturation point, the first and secondtransversal vibration amplitudes b1 and b2 (indicatedin red and blue lines, respectively) jump from thezero response to a finite value. With further increaseof the longitudinal excitation amplitude, the transver-sal vibration amplitudes keep increasing. The energydue to the longitudinal excitation is transferred to thetransversal motion due to the saturation.

Fig. 3 Amplitudes of response b1 as functions of the amplitudeof the excitation; σ1 = 0.005

When there exist multiple stable solutions, a jumpphenomenon associated with varying the excitationamplitude may occur. The trend of amplitude responsesa1, b1, and b2 can be pursued from two ways, i.e., fromf1 = 0 to higher values and vice versa. The jumpphenomena can be chased by tracking the cyan andmagenta arrows, respectively.

We now try to study the effect of internal reso-nance detuning parameter σ2 to the amplitude responsecurves. In Fig. 3, the variation of transversal amplituderesponse b1 has been plotted with respect to the internalresonance detuning parameter σ2. It can be found thatthe increase of detuning parameter from exact internalresonance may postpone the critical excitation valuewhere the saturation phenomenon occurs. Although thephenomenon of saturation is delayed because of theincrease of the detuning parameter, the response ampli-tudes will go higher once the saturation triggered.

The saturation has been discussed in some engi-neering fields, and the readers can refer to [33–35] fordetailed investigation. Since energy transfer is the char-acter of the saturation, this method can be used as vibra-tion suppression technique as presented in Oueini andNayfeh [36].

In Fig. 4, the effect of the external excitation fre-quency detuning parameter to the first-order transversalresponse b1 has been presented. It can be concluded thatthe contribution of the detuning parameter σ1 is similarto that of σ2.

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Fig. 4 Amplitudes of response b1 as functions of the amplitudeof the excitation; σ2 = 0.01

In Fig. 5, the amplitude responses of a1, b1, and b2

are plotted as function of internal resonance detuningparameter σ1 for σ2 = 0, σ2 > 0, and σ2 < 0, respec-tively. The jump phenomenon associated with vary-ing the external excitation frequency is indicated bythe arrows in Fig. 5a. The symmetric behavior of thefrequency–response curve versus external resonancedetuning parameter σ1 can be seen in Fig. 5a wherethe internal resonance is perfectly detuned, i.e., σ2 = 0The nonzero value of σ2 will cause the unsymmetricalconfigurations in frequency–response curves as shownin Fig. 5b, c.

6 Conclusions

The longitudinal and transversal coupling vibrations ofan axially moving beam are considered in this paper. Itis found that by detuning the axially velocity, the exactparameters with which the system present internal reso-nance are located. The Galerkin method is used to trun-cate the governing partial differential equations into aset of coupled nonlinear ordinary differential equations.The method of multiple scales is applied to the govern-ing equations to study the nonlinear dynamics of thesteady-state response caused by the internal–externalresonance.

Examples are presented to show the saturation andjump phenomena when the first longitudinal natural

Fig. 5 Frequency–response curves. a σ2 = 0, b σ2 = 0.02,c σ2 = −0.02

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frequency is equal to the sum of the first two naturalfrequencies of the transversal vibration. The nonlin-ear longitudinal and transversal amplitude–responsecurves with respect to external excitation amplitudeand the external detuning parameters have been inves-tigated for the case that the only longitudinal externalexcitation is implemented. It is concluded that first-order longitudinal external excitation may trigger thetransversal vibration response due to the saturationphenomenon. This phenomenon of saturation has thepotential to be used in the vibration suppression tech-nique.

The effects of the longitudinal external excitationand internal frequency detuning parameters to thetransversal response b1 have been studied. It is foundthat the phenomenon of saturation is delayed due to theincrease of the detuning parameters, while the responseamplitudes obtain higher values once the saturationtriggered for higher value detuning parameters. Thesymmetric and unsymmetrical configurations of theamplitude responses are found as function of internalresonance detuning parameter.

Acknowledgments This investigation is supported by theNational Natural Science Foundation of China (Project Nos.11322214, 11172010, and 11290152) and by the State Key Labo-ratory of Mechanics and Control of Mechanical Structures (Nan-jing University of Aeronautics and Astronautics) under Grant No.MCMS-0112G01.

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