二自由度碰振准哈密顿系统双碰周期解的Melnikov方法
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摘要
采用摄动法和Poincaré映射方法推导出了具有立方非线性项和外部激励项的二自由度碰振系统周期解的扩展Melnikov函数,并运用该Melnikov函数研究了二自由度碰振系统的双碰周期解特性,确定了系统稳定双碰周期2运动的存在条件,即在参数域内的一条临界曲线.通过数值模拟验证,结果表明:该临界曲线下方区域参数是双碰周期2运动,上方区域参数是非双碰周期2运动;当保持其他参数不变,仅增加系统激励幅值f时,系统的运动状态会从多碰多周期运动逐步向双碰周期2运动转变;当保持其他参数不变,仅增加系统恢复系数η0时,系统的运动状态会从双碰周期2运动逐步向多碰多周期运动转变.
Perturbation method and Poincaré mapping method were used to derive the generalized Melnikov function of the periodic solution for a two-degree-of-freedom vibro-impact system with cubic non-linearity and external excitations. By using the Melnikov′s method, the characteristics of periodic motions with double-impact of the2-dof system were studied, and the existence condition of period-2 motions with double-impact was determined as a critical curve in the parameter domain. The results of numerical simulations show that the regions below the critical curve are the period-2 motions with double-impact, the upper regions of the critical curve are not period-2 motions with double-impact;Meanwhile,increasing the force amplitude and keeping the other parameters unchanged, the motion state of the system changes from multi-period motions with multi-impact to period-2 motions with doubleimpact, while increasing the system restitution coefficient and keeping the other parameters unchanged, the motion state of the system changes from period-2 motions with double-impact to multi-period motions with multi-impact.
Perturbation method and Poincaré mapping method were used to derive the generalized Melnikov function of the periodic solution for a two-degree-of-freedom vibro-impact system with cubic non-linearity and external excitations. By using the Melnikov′s method, the characteristics of periodic motions with double-impact of the2-dof system were studied, and the existence condition of period-2 motions with double-impact was determined as a critical curve in the parameter domain. The results of numerical simulations show that the regions below the critical curve are the period-2 motions with double-impact, the upper regions of the critical curve are not period-2 motions with double-impact;Meanwhile,increasing the force amplitude and keeping the other parameters unchanged, the motion state of the system changes from multi-period motions with multi-impact to period-2 motions with doubleimpact, while increasing the system restitution coefficient and keeping the other parameters unchanged, the motion state of the system changes from period-2 motions with double-impact to multi-period motions with multi-impact.
引文
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[19]TIAN R L,ZHAO Z J,YANG X W,et al. Subharmonic bifurcationfor a nonsmooth oscillator[J]. International Journal of Bifurcation&Chaos,2017,27(10):1750163.
[2] SHEN J,LI Y,DU Z. Subharmonic and grazing bifurcations for asimple bilinear oscillator[J]. International Journal of Non-linearMechanics,2014,60(2):70—82.
[3]张思进,杜伟霞,殷珊.振动筛系统双Hopf分岔的反控制[J].湖南大学学报(自然科学版),2017,44(10):55—61.ZHANG S J,DU W X,YIN S. Anti-control of double Hopfbifurcation of vibration rating griddle system[J]. Journal of HunanUniversity(Natural Sciences),2017,44(10):55—61.(In Chinese)
[4] WU Q,COLE C,LUO S,et al. A review of dynamics modeling offriction draft gear[J]. Vehicle System Dynamics,2014,52(6):733—758.
[5]范新秀,王琪.车辆纵向非光滑多体动力学建模与数值算法研究[J].力学学报,2015,47(2):301—309.FAN X X,WANG Q. Research on modeling and simulation oflongitudinal vehicle dynamics based on non-smooth dynamics ofmultibody systems[J]. Chinese Journal of Theoretical&AppliedMechanics,2015,47(2),301—309.(In Chinese)
[6] CAO D,CHU S,LI Z. Study on the non-smooth mechanical modelsand dynamics for space deployable mechanisms[J]. ChineseJournal of Theoretical&Applied Mechanics,2013,45(1):3—15.
[7] KUKUCKA P. Melnikov method for discontinuous planar systems[J]. Nonlinear Analysis Theory Methods&Applications,2007,66(12):2698—2719.
[8] SHEN J,DU Z. Heteroclinic bifurcation in a class of planarpiecewise smooth systems with multiple zones[J]. ZeitschriftAngewandte Mathematik Und Physik,2016,67(3):1—17.
[9] CHáVEZ J P,WIERCIGROCH M. Bifurcation analysis of periodicorbits of a non-smooth Jeffcott rotor model[J]. Communications inNonlinear Science&Numerical Simulation,2013,18(9):2571—2580.
[10] XU J,LI Q,WANG N. Existence and stability of the grazingperiodic trajectory in a two-degree-of-freedom vibro-impactsystem[J]. Applied Mathematics&Computation,2011,217(12):5537—5546.
[11] AL-SHUDEIFAT M A,WIERSCHEM N,QUINN D D,et al.Numerical and experimental investigation of a highly effectivesingle-sided vibro-impact non-linear energy sink for shockmitigation[J]. International Journal of Non-linear Mechanics,2013,52(6):96—109.
[12] LUO G W,ZHU X F,SHI Y Q. Dynamics of a two-degree-offreedom periodically-forced system with a rigid stop:Diversity andevolution of periodic-impact motions[J]. Journal of Sound&Vibration,2015,334(334):338—362.
[13] ZHANG S J,WEN G L,WANG Y. The Melnikov′s method forlocal-subharmonic orbits of a vibro-impact quasi-Hamiltoniansystem[J]. Journal of Vibration Engineering,2016,29(2):214—219.
[14]DU Z,LI Y,SHEN J,et al. Impact oscillators with homoclinic orbittangent to the wall[J]. Physica D Nonlinear Phenomena,2013,245(1):19—33.
[15]YAGASAKI K. Application of the subharmonic Melnikov method topiecewise-smooth systems[J]. Discrete and Continuous DynamicalSystems,2013,33(5):2189—2209.
[16]张思进,文桂林,王紧业,等.碰振准哈密顿系统局部亚谐轨道的Melnikov方法[J].振动工程学报,2016,29(2):214—219.ZHANG S J,WEN G L,WANG J Y,et al. The Melnikov′s methodfor local-subharmonic orbits of a vibro-impact quasi-Hamiltoniansystem[J]. Journal of Vibration Engineering,2016,29(2):214—219.(In Chinese)
[17] XU W,FENG J,RONG H,et al. Melnikov′s method for a generalnonlinear vibro-impact oscillator[J]. Nonlinear Analysis,2009,71(1):418—426.
[18] GRANADOS A,HOGAN S J,SEARA T M. The Melnikov methodand subharmonic orbits in a piecewise smooth system[J]. SiamJournal on Applied Dynamical Systems,2012,11(3):801—830.
[19]TIAN R L,ZHAO Z J,YANG X W,et al. Subharmonic bifurcationfor a nonsmooth oscillator[J]. International Journal of Bifurcation&Chaos,2017,27(10):1750163.