Bifurcation Analysis of a Nonlinear Vibro-Impact System with an Uncertain Parameter via OPA Method

Dongmei Huang,Dang Hong,Wei Li,★,Guidong Yang and Vesna Rajic

1School of Mathematics and Statistics,Xidian University,Xi’an,710071,China

2Department of Mathematics,Changzhi University,Changzhi,046011,China

3Department for Statistics and Mathematics,Faculty of Economics,University of Belgrade,Belgrade,11000,Serbia

ABSTRACT

In this paper,the bifurcation properties of the vibro-impact systems with an uncertain parameter under the impulse and harmonic excitations are investigated.Firstly,by means of the orthogonal polynomial approximation(OPA)method,the nonlinear damping and stiffness are expanded into the linear combination of the state variable.The condition for the appearance of the vibro-impact phenomenon is to be transformed based on the calculation of the mean value.Afterwards,the stochastic vibro-impact system can be turned into an equivalent high-dimensional deterministic non-smooth system.Two different Poincaré sections are chosen to analyze the bifurcation properties and the impact numbers are identified for the periodic response.Consequently,the numerical results verify the effectiveness of the approximation method for analyzing the considered nonlinear system.Furthermore,the bifurcation properties of the system with an uncertain parameter are explored through the high-dimensional deterministic system.It can be found that the excitation frequency can induce period-doubling bifurcation and grazing bifurcation.Increasing the random intensity may result in a diffusion-based trajectory and the impact with the constraint plane,which induces the topological behavior of the non-smooth system to change drastically.It is also found that grazing bifurcation appears in advance with increasing of the random intensity.The stronger impulse force can result in the appearance of the diffusion phenomenon.

KEYWORDS

Orthogonal polynomial approximation;vibro-impact systems;non-smooth systems;grazing bifurcation

1 Introduction

As it is well known,the phenomena of impacts and dry frictions exist widely in a large number of engineering devices [1–3],which can induce the instability and insecurity of these devices.Thus,studying the dynamical properties has been an epoch-making field for solving the relative problems of these non-smooth devices[4–6].In the last few years,numerous papers have concentrated on the study of non-smooth systems[4–8],in particular,on vibro-impact systems.This kind of vibro-impact system appears when a moving mass collides with a barrier and its displacement is greater than a critical value.

Various complex impact structures and models were designed and developed in the past decade;the study of these systems was extensively performed.Namely,since the 1960s,the theoretical and experimental analyses of an impact system under periodic excitation have been performed by Masri[8,9].Based on the local Poincaré mapping method,the stabilities of non-smooth systems have been considered by Nordmark [10],Zhang et al.[11],and Jin et al.[12].Furthermore,the transition phenomena between different bifurcation phenomena of an impact oscillator with viscous damping have been studied by Peterka[13].Luo et al.[14]have researched the bifurcation characteristic of twodegree-of-freedom vibro-impact systems with weak and strong resonance.The global dynamics have been studied in a vibro-impact system with special friction by Gendelman et al.[15].The chaotic attractors and the periodic behavior of an inelastic,forced-impact oscillator near subharmonic resonance conditions have been explored by Rounak et al.[16].Additionally,random factors are unavoidable in the operation of dynamical systems [17,18].By means of the mean Poincaré map,the responses of a vibro-impact system have been analyzed by Feng et al.[19].By using the non-smooth variable transformation in[20],the energy losses induced by the impact and the probability density functions of the impact systems under stochastic excitation have been considered by Dimentberg et al.[21,22].An averaging approach to researching the nonlinear dynamics of the vibro-impact system under the effect of random perturbations has been developed by Namachchivaya et al.[23].Along with researching the multi-valued response of a nonlinear vibro-impact system with the existence of random narrow-band noise [24],Huang et al.have also considered the principal resonances of an elastic impact oscillator under stochastic excitation[25].The stochastic dynamics of the contact force models with elastic impact phenomena under additive noise have been investigated by Kumar et al.[26].By using the traditional theoretical analysis,the stochastic dynamical property of a nonlinear vibroimpact system with Coulomb friction under stochastic noise has been researched by Su et al.[27].Besides investigating the stochastic response of SDOF vibro-impact oscillators under wide-band noise excitations,Qian et al.[28]have also studied the response of vibro-impact systems by the RBF neural network method[29].Although numerous papers have been published,some stochastic non-smooth systems cannot be investigated by these theoretical methods due to the limited application.

In order to overcome the difficulty of studying dynamical systems with an uncertain parameter,the orthogonal polynomial approximation (OPA) method has been utilized to study the stochastic dynamics of some kinds of smooth dynamical systems[30–32]with uncertain parameters.Although different kinds of orthogonal polynomials can be chosen,the Chebyshev polynomial approximation[30,31] and the Laguerre polynomial approximation [33] are the main methods in the stochastic analysis.By using the method based on the derived approximate formula of the Laguerre polynomials,Wang et al.[33]have investigated the stochastic response of a nonlinear elastic impact system,along with studying the global dynamic behavior of this kind of system with an uncertain parameter [34].Utilizing the method of Chebyshev polynomial approximation,Feng et al.[35] have studied the bifurcation of a stochastic nonlinear system with a one-sided constraint.Li et al.[36]have considered the bifurcations of the van der Pol system with two-side barriers.Recently,Huang et al.[37] and Zhang et al.[38]have extended the Chebyshev polynomial approximation method to study nonlinear harvesters.However,so far there have been only few papers on the non-smooth dynamical systems with the OPA method.Therefore,in this paper,we mainly concentrate on studying the stochastic responses of the vibro-impact system with nonlinear stiffness and damping under an uncertain parameter subjected to periodic impulse excitation and harmonic excitation according to the OPA method.

The remaining part of this paper is organized as follows:the equivalent high-deterministic system is derived by the OPA method in Section 2;in Section 3 the effectiveness of the OPA method is proved.Afterwards,the bifurcation phenomena of the system by means of deterministic numerical methods are studied,summarizing the conclusions listed in Section 4.

2 Orthogonal Polynomial Approximation

As for stochastic vibro-impact system with nonlinear stiffness and damping under periodic impulse excitation and harmonic excitation,the non-smooth property is induced by the existence of a rigid barrier,as shown in Fig.1,whose non-dimensional differential equation is given as follows:

wherefcos(ωt) is the harmonic excitation,ais the coefficient of the linear stiffness andcis the coefficient of the nonlinear stiffness.ηis the restitution coefficient(0＜η≤1).Δis the rigid constraint constant.μis an uncertain parameter,denoted asμ=+σξin whichξis a random variable,is the mean value,andσis the intensity ofμ.

When the displacement isx＜Δ,the system(1)is a stochastic smooth nonlinear system as follows:

When the constraint condition isx＜Δand the constraint plane isH={(x,)|x=Δ},the jump equation is=-η,is the velocity before the instant of impacts,andis the velocity after the instant of impacts.

According to the OPA method,the responses of the system(1)without the effect of impact can be expanded into the following sequence:

whereHi(ξ)is theith orthogonal polynomial,herein,the Chebyshev orthogonal polynomials are used[39,40];N=0,1,2,...is the largest number of the expanded terms:xi(t)=∫1-1p(ξ)x(t,ξ)Hi(ξ)dξand

The orthogonality of the polynomials is:

Based on the property of the Chebyshev orthogonal polynomials,the recurrent formula among them is:

As for the case without any constraints,when substituting Eq.(3)into Eq.(2),it has:

By Eq.(5),the cubic terms of Eq.(6) can be expanded into a linear combination of the related single polynomials.The coefficient ofHi(ξ)in the linear combination is denoted asXi(t)andYi(t),(i=0,1,2,...);thus,the nonlinear terms on the left side of Eq.(6)can be expanded into:

The expressions ofXi(t) andYi(t)(i=0,1,2,...) can be derived using software or manual calculation.

Substituting Eqs.(5),(7)and(8)into Eq.(6),the following can be derived:

In order to simplify Eq.(9),both sides of Eq.(9) are multiplied byHi(ξ) and then taken the expectation aboutξby means of the orthogonality of the Chebyshev polynomials,the equivalent deterministic equation without any constraints can be derived:

wherex-1,xN+1,Y-1andYN+1are supposed to be zero due to the approximation in Eq.(3).WhenN→∞in Eq.(3),is strictly equivalent tox(t,ξ);however,in the real calculation,Nis finite and it is also enough to satisfy the basic requirement of the approximation.In the numerical analysis,N=4 will be taken.The expressions ofXi(t),i=0,1,2,3,4,5,which may be used in the discussion,can be found in[37].Additionally,if the derivative ˙xNis denoted asyN,the expressions ofYi(t),i=0,1,2,3,4,5 are given as follows:

Sinceξis a random variable,when a group of sample{ξi}is taken between-1 and 1,each sampleξicorresponds to a deterministic sample system (2); thus,it is necessary to take the mean value to the sample system.The ensemble mean response(EMR)of the stochastic vibro-impact system can be derived by Eq.(3):

By means of Eq.(20),the following relations can be defined:

the mean constraint condition

the mean constraint plane

and the mean jump condition

Substituting Eqs.(9)–(13)and(21)–(23)into the system(1),the stochastic nonlinear vibro-impact system can be simplified as follows:in case ofx＜Δ,

and in case ofx=Δ,the relationship is given as follows:

Eqs.(24)–(28) are the equivalent high-dimensional deterministic systems derived by the mean jump equation,the mean constraint condition,and the Chebyshev polynomial approximation.Consequently,the numerical results of the system with an uncertain parameter can be obtained by effective numerical methods.

3 Bifurcation Analysis

3.1 Deterministic Response

Naturally,in the deterministic case the parametersσ=0.0 orξ=0.0 are satisfied; hence,the original system can be reduced to a deterministic one-side constraint system as follows:

Due to the deterministic property of Eq.(29),the response can be named the deterministic response (DMR).Therefore,for the equivalent system in Eqs.(24)–(28),by setting the parameterσ=0.0,the deterministic response of the high-dimensional system can be derived and denoted by EMR0.Correspondingly,the stochastic response of the equivalent system is denoted as EMR.Thus,the validity of the polynomial approximation for this kind of vibro-impact system with nonlinear stiffness and damping can be verified by comparing EMR0with DMR.The differences between EMR and DMR can show the effect of the uncertain parameter on the responses.

In order to calculate the response,the initial condition of Eq.(29)is chosen asx(0)=-1.0,(0)=0.0.In addition,the initial condition of Eqs.(24)–(28)could be obtained asx(0)=[-1.0,0.0,0.0,0.0],(0)=[0.0,0.0,0.0,0.0].The parameters of the system are given as follows:f=2.0,ω=2.8,a=1.0,=0.8,η=0.8,Δ=0.0,n=2.

Here,two different Poincaré sections(the phase plane and the constraint plane)are taken,while the value ofcvaries between 0.01 and 0.1.The bifurcation diagrams are plotted in Fig.3.It depicts that the properties are the same in Figs.3a and 3b,having been derived based on different Poincaré sections.In order to verify the response results in Fig.3,the phase portraits,the time histories,and the Poincaré sections are exhibited in Fig.4.As Fig.4 displays,whencincreases from 0.035 to 0.05,the 4/4 periodic response of the system converts into the 2/2 response with the asterisk presenting the points of the Poincaré sections.It is also verified the bifurcation properties shown in Fig.3,which are depicted by the points(A1A2A3A4,C1C2C3C4,B1B2,D1D2).In the symbolz=n/p,pis the number of impacts andnis the number of the periods of the excitation force with the periodT=2π/ω.Furthermore,the DMR and EMR0also present very well the consistent base on the phase portraits in Fig.4,showing the availability of the approximation method in Section 2.

Figure 3 :Bifurcation diagrams of DMR and EMR0 vs.c,(a)the phase plane as Poincaré section,(b)the constraint plane as Poincaré section

Figure 4 :(a)The phase portraits and Poincaré sections for c=0.05,(b)the time histories for c=0.05,the periodic 2/2 motion,(c) the phase portraits and Poincaré sections for c=0.035,(d) the time histories for c=0.035,the periodic 4/4 motion

3.2 Effects of an Uncertain Parameter on Period-Doubling Bifurcation

In order to consider the effects of an uncertain parameter on period-doubling bifurcation,the following system parameters are chosen:a=0.5,c=0.05,F0=0.1,f=1.5,u=0.8,η=0.8,Δ=0.0,andn=2.When the excitation frequencyωchanges between 3.98 and 4.2,two examples of numerical results are obtained as displayed in Fig.5 forσ=0.0.Taking the value ofω=3.98,the response characteristics of DMR and EMR0are obtained as shown in Fig.5a.Moreover,with increasingω,DMR and EMR0from the periodic 6/2 motion turn to the approximate 3/1 motion.As for Poincaré section,the mark “■” presents the corresponding Poincaré section,and the periodic properties also can be observed clearly from Poincaré section.

Figure 5 : The phase portraits (a) ω=3.98,the periodic 6/2 motion,(b) ω=4.2,the periodic 3/1 motion

As forω=3.98,when the random intensityσincreases to 0.01,the phase portrait can be seen as plotted in Fig.6a.Even though,the trajectory of the system has certain diffusion,it does not change the property of the motion,the period is still 6T.With the further increase ofσ,EMR in Fig.6b greatly differs from DMR in Fig.5a;actually,it is no longer the periodic motion under the strict sense.For the case ofω=4.2,it also appears to be the same phenomenon,which is plotted in Figs.6c and 6d forσ=0.02 andσ=0.03,respectively.Evidently,the strong diffusion phenomenon results in the change of the topological property of the response in Fig.6d.

As a conclusion,under the effect of the uncertain parameter with the decreasing frequencyωfrom 4.2 to 3.98,the nonlinear vibro-impact system,just like the deterministic system,experiences a period-doubling bifurcation from 3Tto 6T.However,if random intensity continues to increase,the dynamical behavior of the system will experience the essential change.The random factors result in the essential change of the dynamical behavior,the reason of this change is that the trajectories of the system are much closer to the constraint plane; thus,under the random disturbance the trajectories induce diffusion and just impact the constraint plane.

3.3 Existence of Grazing Bifurcation

In this section,to investigate the grazing bifurcation which is typical for the system with impact phenomenon,the same parameters are taken as in Section 3.1,except fora=3.5,f=3.5.

When the frequency isω=4.00495,as presented in Fig.7,one trajectory of the system is tangent to the constraint plane with zero speed,and grazing bifurcation occurs which generally exists only in non-smooth systems.However,in this case with increasing the random intensityσ,the phase portraits undergoes a few changes; as shown in Fig.7.Actually,at some critical points the nonsmooth dynamical response becomes insensitive around the grazing point.Even with relatively large disturbance,the system can also keep the original motion state quiet well(Fig.7b).

At the same time,in case ofω=3.98,the phenomenon of grazing bifurcation also occurs,as shown in Fig.8,but it differs from the case ofω=4.00495 in Fig.7.Whenσ=0.1,the trajectories of the system experience certain diffusion in Fig.8a,and then there is a serious diffusion for biggerσ(as shown in Fig.8b forσ=0.18).This shows that at this grazing bifurcation point,the effect of the uncertain parameter on the response occurs in the same phenomena with period-doubling bifurcation.

Figure 6 : EMR for different values of σ as (a) ω=3.98,σ=0.01,(b) ω=3.98,σ=0.014,(c)ω=4.2,σ=0.02,(d)ω=4.2,σ=0.03

Figure 7 :EMR for different values of σ as ω=4.00495,(a)σ=0.1;(b)σ=0.2

Figure 8 :EMR for different values of σ as ω=3.98,(a)σ=0.1,(b)σ=0.18

In order to present the effect of an uncertain parameter on bifurcation,for different values of intensityσ,the bifurcation diagrams of EMR with the variation ofωare plotted in Figs.9a–9d,respectively.The reasons for the diffusion of the system trajectories are that,on the one hand,the trajectories of the system are much closer to the constraint plane;thus,under the random disturbance the trajectories experience diffusion and then impact the constraint plane.On the other hand,with increasing random intensityσ,the grazing bifurcation appears in advance,which can be seen in Fig.9.

Figure 9 :The bifurcation diagrams of EMR vs.ω for different values of σ,(a)σ=0.0;(b)σ=0.1;(c)σ=0.16;(d)σ=0.18

3.4 Influence of the Impulse Force

When it comes to the influence of the impulse signal on the bifurcation properties the following ideas are considered: the stronger impulse signals are chosen withF0=10.0; the casesn=2 andn=4 are also considered,respectively;and the fact that the different numbers ofnlead to different appearance time and values of impulse.Having compared the bifurcation diagrams plotted in Figs.2 and 10,the following conclusions can be drawn.As it can be seen,at most of the points,the topological properties have not changed,but the diffusion phenomenon apparently has a stronger impulse force,and under larger values ofnthe diffusion phenomenon is more obvious.

Figure 10 : The bifurcation diagrams of EMR0 vs.c with stronger impulse force,(a) the phase plane as Poincaré section with n=2,(b)the constraint plane as Poincaré section with n=2,(c)the phase plane as Poincaré section with n=4,(d)the constraint plane as Poincaré section with n=4

The phase portraits in Fig.11 also verify the results.As illustrated,under smaller impulse force,different values ofnunder the influence of the response are not obvious.With larger value of impulse force and under smaller value ofn,the diffusion phenomenon can be noticed;however,larger value ofncan induce a serious diffusion phenomenon.

4 Conclusions

In this paper,the OPA method is applied in the vibro-impact system with an uncertain parameter under periodic impulse excitation and harmonic excitation,and the damping coefficient is considered as an uncertain parameter.In order to study the dynamical response,the ensemble mean response of an equivalent high-dimensional system is introduced and the impact conditions are also transformed by means of the mean value; then,the deterministic high-dimensional equivalent vibro-impact system is derived.Afterward,the constraint plane and the phase plane are chosen as the Poincaré section,respectively,with the response properties being consistent in the bifurcation diagrams.By combining the analysis of the phase portraits,it is evident that the approximation method is effective in this kind of vibro-impact system.Furthermore,it has been proved that besides period-doubling bifurcation,certain grazing bifurcation exists also in the stochastic nonlinear vibro-impact system.Under the influence of the uncertain parameter,the system responds not only with the characteristic of the smooth system which can make trajectories of the systems’diffusion,but also with a special characteristic of a non-smooth system.At a critical point of bifurcation,the random factors with certain intensity make the dynamical behavior of the system change drastically.Furthermore,grazing bifurcation appears in advance with increasing random intensity.The existence of the stronger impulse force can induce the appearance of a diffusion phenomenon.An appropriate choice of impulse force can control the vibration and improve the response performance.Overall,the bifurcation analysis is helpful for further investigating of stochastic dynamics.

Figure 11 :The phase portraits for c=0.0356,(a)F0=0.1,n=2;(b)F0=0.1,n=4;(c)F0=10.0,n=2;(d)F0=10.0,n=4

Acknowledgement:The authors are grateful for the support by the National Natural Science Foundation of China,the Bilateral Governmental Personnel Exchange Project between China and Slovenia for the Years 2021–2023,Slovenian Research Agency ARRS in Frame of Bilateral Project,the Fundamental Research Funds for the Central Universities,Joint University Education Project between China and East European.

Funding Statement:This work was supported by the National Natural Science Foundation of China(Grant Nos.12172266,12272283),the Bilateral Governmental Personnel Exchange Project between China and Slovenia for the Years 2021–2023 (Grant No.12),Slovenian Research Agency ARRS in Frame of Bilateral Project (Grant No.P2-0137),the Fundamental Research Funds for the Central Universities (Grant No.QTZX23004),Joint University Education Project between China and East European(Grant No.2021122).

Author Contributions:The authors confirm contribution to the paper as follows:Conceptualization,Methodology,Validation:Dongmei Huang;analysis and interpretation of results:Dongmei Huang,Dang Hong,Wei Li;Writing-Original Draft:Dongmei Huang,Dang Hong,Wei Li,Guidong Yang and Vesna Rajic.All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials:Data will be made available on request.

Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.