幾類碰撞振動系統(tǒng)的分岔控制研究
發(fā)布時間：2018-09-05 10:33
【摘要】：碰撞振動是機械工程領域中很普遍的一種現(xiàn)象。一方面,由于碰撞振動系統(tǒng)固有的不連續(xù)特性使系統(tǒng)產生復雜的分岔和混沌等動力學行為,這種非線性行為偏偏又是導致系統(tǒng)失穩(wěn)或結構損壞的原因之一,工程中通常是主動或通過控制迫使系統(tǒng)避開、延遲、或消除這種分岔現(xiàn)象。另一方面,為了某種生產目的,人們開始關注如何主動來利用分岔的非線性特性,通過主動設計或者控制來實現(xiàn)具有所期望特性的分岔。本文以幾類典型的高維碰撞振動系統(tǒng)為研究對象,發(fā)展了相應的控制方法并對碰撞系統(tǒng)的各種余維一分岔、余維二分岔、擦邊非光滑分岔以及一類高維映射退化Neimark-Sacker分岔的控制問題進行了詳細分析并通過實驗調查了一類兩自由度碰撞振動系統(tǒng)豐富的動力學行為。本文主要的研究工作如下:1.研究了慣性式沖擊振動落砂機的擬周期碰撞設計與周期碰撞運動的倍化分岔反控制問題�？紤]到設計過程中經典的Neimark-Sacker分岔臨界準則需要直接計算特征值帶來的局限性,給出了不直接依賴于特征值計算的顯式臨界準則,獲得了系統(tǒng)發(fā)生Neimark-Sacker分岔的兩參數(shù)區(qū)域圖,結合中心流形-范式方法通過選定合適的系統(tǒng)參數(shù)設計出了穩(wěn)定的擬周期碰撞振動。針對慣性式沖擊振動落砂機碰撞的不連續(xù)性和Poincaré映射的隱式特點,在不改變原系統(tǒng)平衡解結構的情況下發(fā)展了一種線性反饋控制方法,利用顯式的周期倍化分岔臨界準則獲得了系統(tǒng)具有較強魯棒性的控制參數(shù)區(qū)域,并應用中心流形-范式方法進一步分析了倍化分岔解的穩(wěn)定性。數(shù)值仿真表明在選定的系統(tǒng)參數(shù)處能設計出穩(wěn)定的擬周期碰撞運動并通過該控制方法實現(xiàn)了落砂機系統(tǒng)的周期倍化分岔。2.研究了一類三自由度含間隙高維雙面碰撞振動系統(tǒng)周期碰撞運動的Neimark-Sacker分岔、Pitchfork分岔以及Hopf-Hopf交互分岔的反控制問題。首先求解得到受控系統(tǒng)的碰撞周期解并建立了六維的Poincaré映射,一般六維映射相應雅克比矩陣的特征值沒有解析的表達式,這使得由特征值特性描述的經典臨界分岔準則在確定控制增益中具有很大的局限性,針對這個局限性給出了六維映射包含特征值分布條件、橫截條件和非共振條件的顯式臨界準則,所建立的準則與經典的分岔準則等價,但并不依賴雅克比矩陣特征值的直接計算,最后基于建立的準則采用反饋控制方法在指定的參數(shù)點實現(xiàn)了高維碰撞系統(tǒng)Poincaré映射Neimark-Sacker分岔、Pitchfork分岔以及Hopf-Hopf交互分岔的反控制。3.研究了一類兩自由度含間隙碰撞振動系統(tǒng)的擦邊分岔并實驗調查了系統(tǒng)的動力學行為。引入不連續(xù)映射推導了系統(tǒng)擦邊附近的范式映射,基于分段的范式映射給出了判別擦邊軌道穩(wěn)定性的條件,數(shù)值揭示了此類碰撞系統(tǒng)不同周期解之間擦邊躍遷的不連續(xù)分岔現(xiàn)象并基于穩(wěn)定性準則進一步驗證了擦邊軌道的穩(wěn)定性。設計并建造了含間隙兩自由度碰撞振動系統(tǒng)的實驗平臺,選取振子和擋板之間不同的間隙距離,通過調節(jié)激振器的激振頻率,實驗揭示了此碰撞振動系統(tǒng)的各種周期運動、擦邊分岔現(xiàn)象和混沌運動的非線性動力學行為。4.研究了一類擴展的Hénon映射退化Neimark-Sacker分岔的反控制問題。利用顯式的Neimark-Sacker分岔臨界準則獲得了線性控制增益的取值區(qū)域,通過中心流形-范式方法將高維映射受控系統(tǒng)簡化為一個二維平面映射,最后利用Chenciner提出的二維平面映射的退化Neimark-Sacker分岔理論設計了多項式函數(shù)非線性反饋控制器,主動實現(xiàn)了系統(tǒng)的退化Neimark-Sacker分岔并數(shù)值仿真驗證了理論分析的正確性。
[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.
【學位授予單位】：湖南大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：TH113.1
[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.
【學位授予單位】：湖南大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：TH113.1
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