Periodic orbits of a conservative 2-DOF vibro-impact system by piecewise continuation: bifurcations and fractals
Periodic orbits of a conservative 2-DOF vibro-impact system by piecewise continuation: bifurcations and fractals
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DOI：
10.1007/s11071-018-04734-4
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发表时间：
2019-01
影响因子：
5.6
通讯作者：
H. Tao;J. Gibert
中科院分区：
文献类型：
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作者：
H. Tao;J. Gibert
用分段连续法研究了一类具有立体机械碰撞模型的保守二自由度振动碰撞系统的精确周期轨道。从放牧解点中提取可行的初始猜测，以便从先前求解的解的每个分支平滑地发起解的每个分支。产生了频率-能量图(FEP)，其中重点放在列举势能分叉上。在已有稳定解分支的多周期副本上发现了额外的临界点，并导致了倍周期分叉的级联。结果表明，系统的完全FEP可以通过向零频率无限分维的过程来观察，在该过程中可以逼近伪周期响应或混沌响应。最后，从数学上和通过FEPS的比较表明，碰撞系统可以显式地表示为具有奇次多项式内力的非线性系统的极值情况。因此，作为线性简正模叠加的对应物，一般保守非线性系统的自由响应可以通过其非线性简正模的分叉来跟踪。
The exact periodic orbits of a conservative 2-degree-of-freedom vibro-impact system with stereo-mechanical impact model is studied using a piecewise continuation method. Feasible initial guesses are extracted from grazing solution points so that each branch of solution can be initiated smoothly from previously solved ones. Frequency-energy plots (FEPs) are produced where an emphasis is placed on enumerating potential bifurcations. Extra critical points are discovered on multiple-period duplicates of existing stable solution branches and lead to cascades of period-multiplying bifurcations. The results indicate that the system’s complete FEP can be viewed through a process of infinite fractals toward zero frequency where pseudo-periodic or chaotic responses are approached. Finally, it is shown both mathematically and through the comparison of FEPs that the impacting system can be represented explicitly as the extreme case of nonlinear systems with an odd-order polynomial internal force. It is thus proposed that as the counterpart to the superposition of linear normal modes, the free responses of a general conservative nonlinear system can be tracked via bifurcations from its nonlinear normal modes.