，，，，*

1.School of Aerospace Engineering，Huazhong University of Science and Technology，Wuhan 430074，P.R.China；

2.Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment，Wuhan 430074，P.R.China

Abstract: The dynamics of an axially accelerating beam subjected to axial flow is studied. Based on the Floquet theory and the Runge-Kutta algorithm，the stability and nonlinear vibration of the beam are analyzed by considering the effects of several system parameters such as the mean speed，flow velocity，axial added mass coefficient，mass ratio，slenderness ratio，tension and viscosity coefficient. Numerical results show that when the pulsation frequency of the axial speed is close to the sum of first- and second-mode frequencies or twice the lowest two natural frequencies，instability with combination or subharmonic resonance would occur. It is found that the beam can undergo the periodic-1 motion under subharmonic resonance and the quasi-periodic motion under combination resonance.With the change of system parameters，the stability boundary may be widened，narrowed or drifted.In addition，the vibration amplitude of the beam under resonance can also be affected by changing the values of system parameters.

Key words：axially accelerating beam；axial flow；subharmonic resonance；combination resonance；Floquet theory

0 Introduction

In the past decades，the dynamics of axially moving beams have received increasing attention due to their widespread industrial applications，such as aerial cables，power transmission belts and band saw blades. Ulsoy et al.［1］，Wickert and Mote［2-3］，and Chen［4］have conducted extensive literature re?views on this dynamical system.

In fact，many real mechanisms can be repre?sented by axially moving beams with pulsating or time-dependent velocities，i.e. axially accelerating beams. The nonlinear dynamical behaviors of axial?ly accelerating beam have been studied extensively for many years.Chen and his coworkers［5-10］conduct?ed many studies on the stability and nonlinear dy?namics of axially accelerating viscoelastic beams by numerical and analytical methods. Sahoo et al.［11-12］discussed the internal resonance，bifurcation and chaotic dynamics of accelerating beams. Based on the method of multiple scales，Wang et al.［13］inves?tigated the principal parametric resonance of an axi?ally accelerating hyperelastic beam. Ghayesh［14］nu?merically calculated the subharmonic dynamics of an axially accelerating beam，showing that the beam can exhibit periodic，quasi-periodic，and chaotic re?sponses with the variation of system parameters such as mean value of the speed. Moreover，some researchers have investigated the nonlinear dynam?ics of axially accelerating structures modeled by Euler?Bernoulli［15］， Rayleigh［16］and Timoshenko beam theories［17］.

In many cases，the surrounding fluid can be ig?nored，as has been done in the aforementioned liter?ature. However，in some special applications such as the steel strip in continuous hot-dip galvanizing process［18-19］and the underwater towed slender struc?tures，the effect of surrounding fluid on the axially moving beam is of great significance. Thus，many investigators have studied the interaction between the axially moving beam and the fluid. Ni et al.［20］and Li et al.［21］respectively studied the stability and dynamics of axially moving cantilevered beam and supported beams in fluid，showing some rich dy?namical behaviors of the beam. Wang and Ni［22］pro?posed a theoretical model of an axially moving beam immersed in fluid supported by two ends with tor?sional springs. Their numerical results showed that the beam can lose stability by buckling.Kheiri et al.［23-24］derived three-dimensional linear equations of motion with the consideration of cross flow effect and stud?ied the dynamics of underwater towed long pipes.They concluded that the pipe may lose stability by either divergence or flutter. Taleb et al.［25］and Gos?selin et al.［26］studied the dynamics and stability of an axially deploying/extruding beam submerged in dense fluid. Motivated by their work，Yan et al.［27］constructed a theoretical model of an extending beam attached to an axially moving base immersed in dense fluid. The numerical results showed that the moving speed of the base can stabilize the beam.Moreover，Ref.［28］and Ref.［29］respectively stud?ied the linear and nonlinear dynamics of an axially sliding cantilevered pipe conveying fluid.In Refs.［18-29］，it was assumed that the beam is immersed in calm fluid，i.e. the axial flow velocity equals to ze?ro. In fact，the case of surrounding fluid flows axial?ly with non-zero velocity is very common in many fields such as ocean engineering. Recently，Yan et al.［30］proposed a simple theoretical model for the dy?namical behavior of an axially moving beam subject?ed to axial flow and derived the nonlinear equation of motion for this system via force balance method.Their numerical results showed that the beam can experience buckling and flutter instabilities with in?creasing axial moving speed. The effects of flow ve?locity，slenderness ratio and some other parameters on the instability mode，buckling displacement and flutter amplitude of the beam were explored. In their study，however，they only considered the beam moving with a constant axial speed. In practice，the beam may undergo axially accelerating or decelerat?ing motions in many cases. Motivated by this，we will further expand the existing works by consider?ing an axially accelerating beam with time-varying moving speed.

In this paper，a theoretical model of an axially accelerating beam supported at both ends and sub?jected to axial flow is established. The nonlinear equation of motion is derived first and then dis?cretized into a set of nonlinear ordinary differential equations via the Galerkin’s technique. Based on the Floquet theory and the Runge-Kutta algorithm，the stability and dynamic response of the beam are ob?tained，and the effects of axially moving speed，flow velocity and several other system parameters on the dynamical behaviors of the beam are analyzed.

1 Problem Formulation

Fig.1 shows a simply-supported beam of lengthl，diameterD，area moment of inertiaI，and mass per unit lengthmtraveling at a time-dependent axial speedV（t）under an applied tensionN0. It is as?sumed that the beam is made of viscoelastic material of the Kelvin-Voigt model and hence the flexural stiffness of the beam may be written asE0I(1+γ?/?t)，withE0andγbeing the Young’s modulus and viscoelastic coefficient respectively. In addition，the axially accelerating beam is subjected to an axial flow with densityρa(bǔ)nd velocityVf.

The nonlinear equation of motion of the axially moving beam in axial flow has been derived previ?ously by Yan et al.［30］，and can be given by the fol?lowing dimensionless form

In Eq.（1），several dimensionless quantities and parameters are defined by

wherew(x，t) andη(ξ，τ) are the dimensional and dimensionless transverse displacement of the beam；the over-dot and the prime denote the derivatives with respect to dimensionless timeτand the coordi?nate of the centerline of the beamξ，respectively；vandvfare the dimensionless axial speed of the beam and the flow velocity，respectively；φandφare two kinds of mass ratio；βandεare the axial added mass coefficient and the slenderness ratio，respectively；cfandcdare the frictional and the form drag coeffi?cients，respectively；andΓare the dimensionless viscosity coefficient and tension， respectively；sgn（vf-v） in Eq.（1） is a sign function，i. e.sgn（vf-v）=1 ifvf＞v；sgn（vf-v）=-1 ifvf＜v，and sgn（vf-v）=0 ifvf=v. For an accelerating beam，the axial speedvis characterized as a small periodic pulsation on the mean speedv0，namely

whereaandωare the dimensionless pulsating ampli?tude and frequency，respectively.

2 Solution Method

The governing equation can be discretized by applying Galerkin’s technique，with the simply-sup?ported beam eigenfunctionssin(jπξ)being the admissible functions，thus

whereqj(τ) is the corresponding generalized coordi?nates.

Substituting the expression of Eq.（4） into Eq.（1），multiplying byφi(ξ) and integrating from 0 to 1 leads to

whereM，C，KandNrepresent the structural mass matrix，damping matrix，stiffness matrix and non?linear vector，respectively. The elements of these matrices are given by

where

withδijbeing the Kronecker delta function.

Due to the fact that some coefficients of Eq.（5）are time-dependent and periodic，the Floquet theory can be utilized for stability analysis［31］. The Runge-Kutta algorithm will be used to solve Eq.（5）for nonlinear dynamics analysis. Throughout，unless otherwise specified，a truncation ofN=4 in the Galerkin’s method will be chosen for numerical cal?culations. The convergence test of theN=4 trunca?tion is carried out and the bifurcation diagrams are shown in Figs.2（a，b）forN=2，3 and 4. As can be observed that，N=4 is an optimal choice，for the cases ofvf=3，v0=1 andvf=5，v0=1.

Fig.2 Bifurcation diagrams of beam’s responses by using different N

According to Yan et al.［30］，several system pa?rameters in the numerical calculation are given by

For the sake of simplicity，the over-bar ofwill not be shown in the following analysis.

3 Results and Discussion

3.1 Stability analysis

In this subsection，F(xiàn)loquet theory is applied to study the stability of the axially accelerating beam system. The effects of several system parameters such as mean speedv0，flow velocityvf，axial added mass coefficientβ，slenderness ratioε，mass ratioφ，tensionΓand viscoelasticity coefficientγon the stability boundary are analyzed.

Consider an axially accelerating beam subjected to an axial flow withvf=5. The stability boundaries in plane（ω，a）are shown in Fig.3 forv0=0.5，1 and 1.5. The first two natural frequencies of the beam are calculated by an eigenvalue analysis and given in Table 1. By inspecting the results shown in Fig.3 and Table 1，subharmonic and combination resonances of the first and second modes can be ob?served in the vicinity of 2ω1，2ω2andω1+ω2in the case ofv0=1.5. However，the combination reso?nance and the subharmonic resonance of the first mode disappear asv0decreases from 1 to 0.5. More?over，the decrease ofv0makes the stability boundar?ies move towards the increasing direction of the pul?sating amplitudea，and drift along the positive direc?tion of the pulsating frequencyωin plane（ω，a），which makes the unstable region become narrow. In other words，the smaller mean speedv0leads to the lager instability threshold ofafor a givenω，and the smaller unstable range ofωfor a givena.Thus，one can conclude that the increase in mean speedv0makes the beam system more prone to instability.

Fig.3 Stability boundaries for various v0(vf=5)

Table 1 Natural frequencies of beam for various v0(vf=5)

Recalling the previous work of Ref.［30］，where the problem of an axially moving beam with constant speed in axial flow was considered，it was shown that the beam would lose stability at a critical axial speedvcr=3.05 asvf=5（Fig.3 in Ref.［30］）.In this study，however，it can be seen from Fig.3 that the axially accelerating beam withvf=5 has the possibil?ity of losing stability via parametric resonance when the axial speed“v”is within the range of［0.5，1.5］fora=0.5 andv0=1，or within the range of［0.2，0.8］fora=0.6 andv0=0.5. It is obvious that under the same other system parameters，the minimum moving speed for instability of the axially accelerat?ing beam is much lower than that of a beam with con?stant axially moving speed in Ref.［30］.

The influences ofvf，β，ε，φ，Γandγon the stability boundaries are presented in Fig.4. As a sup?plement，Tables 2—7 give the natural frequencies of the beam for several typical cases.

Table 2 Natural frequencies for beam with different vf(v0=1)

Table 3 Natural frequencies for beam with different β(vf=3，v0=1)

Table 4 Natural frequencies for beam with different ε(vf=3，v0=1)

Table 5 Natural frequencies for beam with different φ(vf=3，v0=1)

Table 6 Natural frequencies for beam with different Γ(vf=3，v0=1)

Table 7 Natural frequencies for beam with different γ(vf=3，v0=1)

The stability boundaries in plane（ω，a）are shown in Fig.4（a）forvf=0.2，3，5 andv0=1. In this case，the subharmonic and combination reso?nances of the first and second modes are observed asvf=0.2 and 3. However，the combination resonance vanishes asvfincreases to 5. In addition，one can al?so find that the increase ofvfmakes the stability boundaries move towards the increasing direction ofaand drift slightly along the positive direction ofωin plane（ω，a），which makes the unstable region be?come narrow. By comparing Fig.3 with Fig.4（a），one can find that thev0andvfhave opposite effects on the stability of the beam.

Fig.4 Stability boundaries of beam with different parameters

For fixed values of mean axial speed and flow velocity，F(xiàn)igs.4（b—f）show the stability boundar?ies of the beam as several parameters are varied，forvf=3 andv0=1. The subharmonic and combination resonances of the first and second modes can be ob?served. By comparing Figs.4（b—d），it is noted that slenderness ratioεand mass ratioφhave the same effect on the stability boundaries asvf. However，the influence of axial added mass coefficientβon the stability boundaries is similar to that ofv0，namely，the decreasing ofβmakes the stability boundaries moves towards the increasing direction ofaand drift along the negative direction ofωin plane（ω，a）and the unstable region becomes narrow.

From Fig.4（e），it is found that the effect of tensionΓis mainly reflected in making the unstable region drift to the left whenΓis decreased. That means that the increase ofΓwould make the stabili?ty boundaries move towards the increasing direction ofω. However，the size of the unstable region in plane（ω，a）does not change much. On the con?trary，the viscoelasticity coefficientγcan change the size of the unstable region in plane（ω，a），while it has little effect on the shift of the stability boundar?ies，as can be seen from Fig.4（f）. It is also noted that the presence ofγcan enhance the stability of the beam system.

What’s more，these figures shown above indi?cate that the stability boundary for the summation resonance is most sensitive to the change of all sys?tem parameters discussed.

3.2 Nonlinear dynamic analysis

According to the linear stability analysis，it is noted that the beam can experience subharmonic and combination resonances asaandωvary. To fur?ther understand the dynamical behaviors of the beam at resonance，the vibration responses of the beam at resonance are investigated by a nonlinear dynamic analysis in this subsection.

In the nonlinear dynamic analysis，attention is concentrated on the vibrations of the midpoint of the beam（ξ=0.5）. Effects of system parameters on the vibration responses of the beam can be summa?rized in the form of bifurcation diagrams，by record?ing the amplitude of the beam whenever the vibra?tion velocity atξ=0.5 becomes zero.

Fig.5 shows several bifurcation diagrams of the beam’s responses for different values of mean speedv0and pulsating amplitudea. As can be seen from Fig.5（a），forvf=5 anda=5，the beam remains stable within theωrange of［0，100］forv0=0.5.However，the beam would lose stability in someωranges asv0increases to 1 or 1.5. Recalling the sta?bility boundaries shown in Fig.3，it can be seen that whenv0=1.5，the beam suffers，respectively，a first-mode subharmonic resonance，a combination resonance and a second-mode subharmonic reso?nance in theωranges of［16.4，21.6］，［47.8，50.2］and［70.5，90.4］. Whenv0=1，the first-mode and second-mode subharmonic resonances occur，re?spectively in the ω ranges of ［20.4，22.6］ and［76，88.2］.

In order to analyze the vibration mechanism of the beam at resonance，phase portraits，Poincaré maps and power-spectrum-density（PSD）diagrams are utilized here as powerful techniques in distin?guishing chaotic responses from periodic or quasi-pe?riodic motions.

Forvf=5 andv0=1.5，the cases ofω=20 and 49 are chosen as two typical samples for the beam at subharmonic and combination resonances.In the case ofω=20，the phase portrait shown in Fig.6（a1）presents only a limit cycle，the Poin?caré map shows a pair of symmetrical points（Fig.6（a2）），and the PSD curve is clear with several obvious peaks and a limited frequency bandwidth （Fig.6（a3））. These results indicate that the beam undergoes a periodic motion. Ac?cording to Pa?doussis［32］，the Poincaré map con?sists of a number of points equal to twice the peri?od number when the motion is periodic. Thus，the beam definitely undergoes a periodic-1 motion whenω=20. As for the case ofω=49，one can find that a limited number of cycles are contained in the phase portrait（Fig.6（b1）），and a series of points forming a circle are displayed in the Poin?caré map（Fig.6（b2））. By inspecting Fig.6（b3）further，the PSD curve has several peaks and a broader frequency bandwidth. Thus，we can point out that the beam undergoes a quasi-periodic mo?tion whenω=49.

Compared with a beam with constant axially moving speed，once the axially accelerating beam becomes unstable，oscillation rather than statically buckling would occur. This is because that paramet?ric resonance is the preferred form of instability of the axially accelerating beam，showing the most im?portant difference between the dynamical system and that of Ref.［30］.

Indeed，based on more extensive calculations，it is found that the phase portraits，Poincaré maps and PSD diagrams for several other pulsating fre?quencies in the resonance ranges are similar to the results of Fig.6. Thus we can conclude that the beam would undergo periodic-1 and quasi-periodic motions in subharmonic resonance and the combina?tion resonance，respectively.

A slightly different bifurcation diagram of Fig.5（b）is constructed for the beam withvf=3，v0=1 anda=0.4，0.5，0.6. Results show that the beam loses stability in theωranges of［18.5，21］and［74.9，84.8］fora=0.4；in theωranges of［18，21.5］and［73.2，86.8］fora=0.5；in theωranges of［17.6，21.9］，［48.6，50.2］and［73.2，86.8］fora=0.5. Theseωranges for unstable behavior of the beam are consistent with the results in the linear sta?bility analysis.

Fig.5 Bifurcation diagrams of beam’s responses

From Fig.5，one can also find that the overall vibration amplitude of the beam at resonance in?creases with the increase ofv0，aandω.

Effects ofvf，β，ε，φ，Γandγon the vibra?tion response of the beam are summarized via bifur?cation diagrams presented in Fig.7. It should be not?ed that all the values of system parameters are cho?sen to be the same as those defined in Fig.4. Due to the fact that the effects of these parameters on theωranges for the beam’s instability are consistent with that obtained in the linear stability analysis，only the vibration amplitudes of the beam will be focused on. Figs.7（c，d）show the bifurcation diagrams of the beam for different values ofεandφ，respective?ly. Both figures show that the larger the values of these two parameters are，the smaller the ampli?tudes of the beam at resonance become. It can be found from Fig.7（b）that increasingβcan increase the vibration amplitude of the beam. And the in?crease ofΓcan increase the vibration amplitude of the beam at the first-mode subharmonic resonance and slightly reduce the amplitude of the beam at the second-mode subharmonic resonance，which can be observed in Fig.7（e）. Comparing Figs.7（a，f），it is noted thatvfandγhave the same effect on the vi?bration amplitude，i.e.，increasingvforγcan re?duce the amplitude of the beam at the first-mode subharmonic resonance and accelerate the growth rate of the vibration amplitude of the beam with in?creasingωin the second-mode subharmonic reso?nance region.

Fig.6 Phase portraits, Poincaré maps and PSD diagrams for vf= 5 and v0=1.5

Fig.7 Bifurcation diagrams of beam’s responses with different system parameters

4 Conclusions

A theoretical analysis of axially accelerating beams in axial flow has been conducted. The stabili?ty and nonlinear dynamic analyses are conducted by Floquet theory and Runge-Kutta algorithm，respec?tively. Extensive numerical calculations are conduct?ed to analyze the effects of several system parame?ters such as mean axial speed，flow velocity，axial added mass coefficient，mass ratio，slenderness ra?tio，tension and viscosity coefficient on the stability and vibration response of the beam.

In the stability analysis，results show that the beam can occur first-mode and second-mode subhar?monic resonances in the vicinity of twice the lowest two natural frequencies respectively，and combina?tion resonance in the vicinity of the sum of first- and second-mode frequencies. Effects of several system parameters on the stability boundaries can be sum?marized as follows：

（1）Increasing the mean speed and axial added mass coefficient can widen the instability regions.

（2）The increase of flow velocity，slenderness ratio，mass ratio，tension and viscoelasticity coeffi?cient can narrow the instability regions.

（3）The stability boundaries can shift along the axis of pulsating frequency as some parameters（ex?cept viscoelasticity coefficient）vary.

（4）The stability boundary for the summation resonance is most sensitive to the change of all sys?tem parameters.

In the nonlinear dynamic analysis，the conclu?sion can be drawn out that the beam undergoes peri?odic-1 and quasi-periodic motions at subharmonic resonance and combination resonance，respectively.Results also show that the vibration amplitude of the beam at resonance can be affected by several key system parameters. Generally speaking，the vibra?tion amplitude would increase with the increase of axial added mass coefficient and pulsating frequency but decreases with the increase of slenderness ratio and mass ratio. However，the increase of flow ve?locity and viscoelasticity coefficient can reduce the amplitude of the beam at the first-mode subharmon?ic resonance. Moreover，increasing the tension can increase the vibration amplitude of the beam at the first-mode subharmonic resonance and slightly re?duce the amplitude of the beam at the second-mode subharmonic resonance.

AcknowledgementsThis work was supported by the Na?tional Natural Science Foundation of China （Nos.11972167，12072119 and 12102139）.

AuthorsDr. YAN Hao received the Ph.D. degree in solid mechanics at Huazhong University of Science and Technolo?gy，Wuhan，China，in 2019. He is currently a postdoc of School of Aerospace Engineering，Huazhong University of Science and Technology，Wuhan，China. His research is fo?cused on pipe conveying fluid，axially moving structures and nonlinear dynamics.

Prof. WANG Lin is currently a full professor of School of Aerospace Engineering，Huazhong University of Science and Technology，Wuhan，China. His research has focused on dynamics and control of fluid conveying pipe，fluid-struc?ture interaction and aeroelasticity of aircraft structure.

Author contributionsDr. YAN Hao constructed the mod?el, conducted the analysis, interpreted the results and wrote original manuscript. Prof. NI Qiao reviewed the writing.Prof. WANG Lin supervised this study, polished English writing of the manuscript and contributed to the discussion and revision of the study. Mr. DAI Huliang helped perform the analysis with constructive discussions. Dr. ZHOU Kun contributed to numerical analysis. All authors commented on the manuscript draft and approved the submission.

Competing interestsThe authors declare no competing interests.