几类碰撞振动系统的分岔控制研究

发布时间：2018-09-05 10:33

【摘要】：碰撞振动是机械工程领域中很普遍的一种现象。一方面,由于碰撞振动系统固有的不连续特性使系统产生复杂的分岔和混沌等动力学行为,这种非线性行为偏偏又是导致系统失稳或结构损坏的原因之一,工程中通常是主动或通过控制迫使系统避开、延迟、或消除这种分岔现象。另一方面,为了某种生产目的,人们开始关注如何主动来利用分岔的非线性特性,通过主动设计或者控制来实现具有所期望特性的分岔。本文以几类典型的高维碰撞振动系统为研究对象,发展了相应的控制方法并对碰撞系统的各种余维一分岔、余维二分岔、擦边非光滑分岔以及一类高维映射退化Neimark-Sacker分岔的控制问题进行了详细分析并通过实验调查了一类两自由度碰撞振动系统丰富的动力学行为。本文主要的研究工作如下:1.研究了惯性式冲击振动落砂机的拟周期碰撞设计与周期碰撞运动的倍化分岔反控制问题。考虑到设计过程中经典的Neimark-Sacker分岔临界准则需要直接计算特征值带来的局限性,给出了不直接依赖于特征值计算的显式临界准则,获得了系统发生Neimark-Sacker分岔的两参数区域图,结合中心流形-范式方法通过选定合适的系统参数设计出了稳定的拟周期碰撞振动。针对惯性式冲击振动落砂机碰撞的不连续性和Poincaré映射的隐式特点,在不改变原系统平衡解结构的情况下发展了一种线性反馈控制方法,利用显式的周期倍化分岔临界准则获得了系统具有较强鲁棒性的控制参数区域,并应用中心流形-范式方法进一步分析了倍化分岔解的稳定性。数值仿真表明在选定的系统参数处能设计出稳定的拟周期碰撞运动并通过该控制方法实现了落砂机系统的周期倍化分岔。2.研究了一类三自由度含间隙高维双面碰撞振动系统周期碰撞运动的Neimark-Sacker分岔、Pitchfork分岔以及Hopf-Hopf交互分岔的反控制问题。首先求解得到受控系统的碰撞周期解并建立了六维的Poincaré映射,一般六维映射相应雅克比矩阵的特征值没有解析的表达式,这使得由特征值特性描述的经典临界分岔准则在确定控制增益中具有很大的局限性,针对这个局限性给出了六维映射包含特征值分布条件、横截条件和非共振条件的显式临界准则,所建立的准则与经典的分岔准则等价,但并不依赖雅克比矩阵特征值的直接计算,最后基于建立的准则采用反馈控制方法在指定的参数点实现了高维碰撞系统Poincaré映射Neimark-Sacker分岔、Pitchfork分岔以及Hopf-Hopf交互分岔的反控制。3.研究了一类两自由度含间隙碰撞振动系统的擦边分岔并实验调查了系统的动力学行为。引入不连续映射推导了系统擦边附近的范式映射,基于分段的范式映射给出了判别擦边轨道稳定性的条件,数值揭示了此类碰撞系统不同周期解之间擦边跃迁的不连续分岔现象并基于稳定性准则进一步验证了擦边轨道的稳定性。设计并建造了含间隙两自由度碰撞振动系统的实验平台,选取振子和挡板之间不同的间隙距离,通过调节激振器的激振频率,实验揭示了此碰撞振动系统的各种周期运动、擦边分岔现象和混沌运动的非线性动力学行为。4.研究了一类扩展的Hénon映射退化Neimark-Sacker分岔的反控制问题。利用显式的Neimark-Sacker分岔临界准则获得了线性控制增益的取值区域,通过中心流形-范式方法将高维映射受控系统简化为一个二维平面映射,最后利用Chenciner提出的二维平面映射的退化Neimark-Sacker分岔理论设计了多项式函数非线性反馈控制器,主动实现了系统的退化Neimark-Sacker分岔并数值仿真验证了理论分析的正确性。
[Abstract]:Collision vibration is a very common phenomenon in the field of mechanical engineering. On the one hand, due to the inherent discontinuity of the collision vibration system, complex bifurcation and chaos and other dynamic behaviors occur in the system. This nonlinear behavior bias is also one of the reasons leading to the instability or structural damage of the system. In engineering, it is usually active or through control. Force the system to avoid, delay, or eliminate this bifurcation phenomenon. On the other hand, for some production purposes, people begin to pay attention to how to actively utilize the nonlinear characteristics of the bifurcation to achieve the desired bifurcation characteristics by active design or control. Corresponding control methods are analyzed in detail for various kinds of codimension bifurcations, codimension bifurcations, edge-rubbing nonsmooth bifurcations and degenerate Neimark-Sacker bifurcations of a class of two-degree-of-freedom impact vibration systems. In this paper, the quasi-periodic collision design and bifurcation inverse control of periodic collision motion of an inertial impact-vibration blasting machine are studied. A two-parameter domain diagram of Neimark-Sacker bifurcation is obtained, and a stable quasi-periodic impact vibration is designed by choosing the appropriate system parameters with the central manifold-normal method. In this paper, a linear feedback control method is developed. The robust control parameter region of the system is obtained by using the explicit periodic doubling bifurcation critical criterion, and the stability of the doubling bifurcation solution is further analyzed by using the central manifold-normal method. The quasi-periodic collision motion of a three-degree-of-freedom high-dimensional two-sided collision vibration system with clearance is studied. The inverse control problems of the Neimark-Sacker bifurcation, the Pitchfork bifurcation and the Hopf-Hopf interacting bifurcation of the periodic collision motion are studied. The periodic solution of collision and the six-dimensional Poincare mapping are established. Generally, the eigenvalues of the corresponding Jacobian matrices of the six-dimensional mapping have no analytic expression. This makes the classical critical bifurcation criterion described by the eigenvalue properties have great limitation in determining the control gain. To overcome this limitation, the eigenvalue fraction of the six-dimensional mapping is given. The explicit critical criteria for the distribution condition, transversal condition and non-resonance condition are equivalent to the classical bifurcation criteria, but do not depend on the direct calculation of the eigenvalues of Jacobian matrices. Finally, based on the established criteria, the Poincar maps Neimark-Sacker scores of high-dimensional collision systems are achieved at specified parameters by using feedback control method. Bifurcation, Pitchfork bifurcation and Hopf-Hopf interacting bifurcation anti-control are studied. 3. Boundary-rubbing bifurcation of a two-degree-of-freedom vibration system with clearance impact is studied and its dynamic behavior is investigated experimentally. Based on the stability criterion, the stability of the rubbed track is further verified. An experimental platform of a two-degree-of-freedom impact vibration system with clearances is designed and constructed. The different clearance distances between the oscillator and the baffle plate are selected and the stability of the rubbed track is further verified. By adjusting the excitation frequency of the exciter, the experimental results reveal various periodic motions, edge-rubbing bifurcation phenomena and nonlinear dynamical behaviors of chaotic motions of the impact vibration system. 4. The anti-control problem of degenerate Neimark-Sacker bifurcation for a class of extended H_ non maps is studied. The linear control is obtained by using the explicit Neimark-Sacker bifurcation critical criterion. The control system of high-dimensional mapping is simplified to a two-dimensional planar mapping by the central manifold-normal method. Finally, a polynomial function nonlinear feedback controller is designed by using the degenerate Neimark-Sacker bifurcation theory of two-dimensional planar mapping proposed by Chenciner, and the degenerate Neimark-Sacker bifurcation of the system is realized actively. The correctness of theoretical analysis is verified by numerical simulation.
【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：TH113.1

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