带有时滞位移控制的轴向行进弦横向振动响应
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摘要
多种工程系统如动力传送带、纺织纤维、空中缆车索道和高空升降机等均涉及轴向运动弦线的横向振动问题。轴向行进弦的横向振动不仅会降低结构的使用寿命,而且还可能造成灾难性后果,所以对横向振动进行控制是非常有必要的。然而对振动进行控制的过程中不可避免的存在着时滞,一方面,这些时滞可能引起系统动力学行为的定性改变,如影响系统稳定性,产生复杂的动力学响应等；另一方面,可将时滞作为反馈控制参数,应用到轴向行进弦横向振动的控制中。
本文引入共置的传感器和激励机,对直接时滞位移反馈控制器作用下轴向行进弦的横向振动问题进行了研究。主要工作包括：
(1)建立了时滞位移反馈控制作用下轴向行进弦横向非线性振动的动力学模型,通过二阶伽辽金离散方法得到了行进弦受控系统的微分差分型泛函方程组。
(2)将Belair定理推广到了N维系统,证明了对于含有时滞的指数多项式形式的超越特征值方程,当时滞连续变化时,只有当特征值穿越虚轴时特征值实部大于零的数目才会发生变化,并指出轴向行进弦的平衡点在时滞位移控制器作用下会通过Hopf分岔发生失稳。确定了平衡点在时滞和反馈位移增益参数域内的稳定性划分,发现存在多个稳定的参数区域。采用数值积分方法研究了由时滞引起的系统稳态响应多个吸引子共存现象。
(3)对于受非线性时滞位移反馈和简谐外激励作用的行进弦系统,应用泛函分析和中心流形约化的方法,重点研究了行进弦在单Hopf分岔点附近的局部动力学行为。将行进弦二阶截断系统同调为中心流形上单复变量的常微分方程,由平均法给出周期解的近似解析形式,并对周期解的稳定性进行了判定。数值仿真表明,在Hopf分岔点的邻域内,近似解析解和数值结果有很好的一致性。
(4)最后,利用Poincare映射讨论了时滞对周期解分岔行为的影响,发现了时滞引起的准周期解共存现象。
Transverse vibration of axially moving strings is involved in many engineering devices such as power transmission belts, thread lines, aerial cable tramways and aether lifts. It is shown that time delay is inevitably in controllers. And the time delay often affects the dynamics of systems essentially which can not only affect the stability of systems, but also lead to the complex dynamics of systems. However, an artificially introduced delay in the feedback can play an essential role in stabilizing the transverse vibration of axially moving strings.
The local dynamics of an axially moving string under aerodynamic forces are investigated with a time-delayed position feedback controller. The research work can be summarized as follows:
(1) The dynamical model of transverse vibration of axially moving strings with a time-delayed position feedback controller is established. The difference-differential governing equation is obtained in modal coordinates of a two-degree-of-freedom system through the Galerkin's discrete procedure.
(2) The Belair Theorem is advanced to a more generalized theorem for any polynomial-exponential equations with constant time delay. It is proved that as the time delay varies, the number of solutions of the characteristic equation can only be changed when the eigenvalue passes through the imaginary axis. The Hopf bifurcation curves are presented in the space of controlling parameter. Two different kinds of periodic solutions are reported.
(3) A new delayed system is obtained by adding nonlinear delayed position feedback and external excitation to the original system. With the aid of the center manifold reduction, a functional analysis is carried out to reduce the modal equation to a single ordinary differential equation in one complex variable on the center manifold. The approximate analytical solutions in the vicinity of Hopf bifurcations are derived in the case of primary resonance. A periodic solution expressed in the closed form is found to be in good agreement with that obtained by numerical simulation. A Poincare section is defined to find the stability of periodic solutions. Two different kinds of quasi-periodic solutions are reported.
本文引入共置的传感器和激励机,对直接时滞位移反馈控制器作用下轴向行进弦的横向振动问题进行了研究。主要工作包括：
(1)建立了时滞位移反馈控制作用下轴向行进弦横向非线性振动的动力学模型,通过二阶伽辽金离散方法得到了行进弦受控系统的微分差分型泛函方程组。
(2)将Belair定理推广到了N维系统,证明了对于含有时滞的指数多项式形式的超越特征值方程,当时滞连续变化时,只有当特征值穿越虚轴时特征值实部大于零的数目才会发生变化,并指出轴向行进弦的平衡点在时滞位移控制器作用下会通过Hopf分岔发生失稳。确定了平衡点在时滞和反馈位移增益参数域内的稳定性划分,发现存在多个稳定的参数区域。采用数值积分方法研究了由时滞引起的系统稳态响应多个吸引子共存现象。
(3)对于受非线性时滞位移反馈和简谐外激励作用的行进弦系统,应用泛函分析和中心流形约化的方法,重点研究了行进弦在单Hopf分岔点附近的局部动力学行为。将行进弦二阶截断系统同调为中心流形上单复变量的常微分方程,由平均法给出周期解的近似解析形式,并对周期解的稳定性进行了判定。数值仿真表明,在Hopf分岔点的邻域内,近似解析解和数值结果有很好的一致性。
(4)最后,利用Poincare映射讨论了时滞对周期解分岔行为的影响,发现了时滞引起的准周期解共存现象。
Transverse vibration of axially moving strings is involved in many engineering devices such as power transmission belts, thread lines, aerial cable tramways and aether lifts. It is shown that time delay is inevitably in controllers. And the time delay often affects the dynamics of systems essentially which can not only affect the stability of systems, but also lead to the complex dynamics of systems. However, an artificially introduced delay in the feedback can play an essential role in stabilizing the transverse vibration of axially moving strings.
The local dynamics of an axially moving string under aerodynamic forces are investigated with a time-delayed position feedback controller. The research work can be summarized as follows:
(1) The dynamical model of transverse vibration of axially moving strings with a time-delayed position feedback controller is established. The difference-differential governing equation is obtained in modal coordinates of a two-degree-of-freedom system through the Galerkin's discrete procedure.
(2) The Belair Theorem is advanced to a more generalized theorem for any polynomial-exponential equations with constant time delay. It is proved that as the time delay varies, the number of solutions of the characteristic equation can only be changed when the eigenvalue passes through the imaginary axis. The Hopf bifurcation curves are presented in the space of controlling parameter. Two different kinds of periodic solutions are reported.
(3) A new delayed system is obtained by adding nonlinear delayed position feedback and external excitation to the original system. With the aid of the center manifold reduction, a functional analysis is carried out to reduce the modal equation to a single ordinary differential equation in one complex variable on the center manifold. The approximate analytical solutions in the vicinity of Hopf bifurcations are derived in the case of primary resonance. A periodic solution expressed in the closed form is found to be in good agreement with that obtained by numerical simulation. A Poincare section is defined to find the stability of periodic solutions. Two different kinds of quasi-periodic solutions are reported.
引文
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[2]CHEN L Q. Analysis and control of transverse vibrations of axially moving strings [J]. Applied Mechanics Reviews,2005,58:91-116.
[3]张伟.轴向行进弦线横向振动的控制[D].上海大学博士学位论文,2006.4.
[4]SKUTCH R. Uber die Bewegung Eines Gespannten Fadens Weicher Gezwungun ist Durch Zwei Feste Lpunkte mit Einer Constantan Geschwindigkeit zu gehen und Zwischen denselben in Transversal Schwingungen von gerlinger Amplitude Versetzi Wird [J]. Annalen der Physik und Chenmie,1897,61:190-195.
[5]MOTE C D JR. Dynamic stability of axially moving materials [J]. The Shock and Vibration Digest,1972,4(4):2-11.
[6]WICKERT J A, MOTE C D JR. Current research on the vibration and stability of axially moving materials [J]. The Shock and Vibration Digest,1988,20(5):3-13.
[7]WANG K W, LIU S P. On the noise and vibration of chain drive systems [J]. The Shock and Vibration Digest,1991,23(4):8-13.
[8]ABRATE A S. Vibration of belts and belt drives [J]. Mechanism and Machine Theory,1992, 27(6):645-659.
[9]MEIROVITCH L. Dynamics and Control of Structures [M]. New York:Wiley,1990.
[10]HUGHES P C, SKELTON R E. Controllability and observability of linear matrix-second-order systems [J]. Journal of Applied Mechanics,1980,47(2):415-420.
[11]YANG B, MOTE C D JR. Controllability and observability of distributed gyroscopic systems [J]. Journal of Dynamic Systems, Measurement, and Control,1991,113(1):11-17.
[12]KOSTYUK V I, KRASNOPROSHINA A A, ILYUKHIN A G. Vibration of a longitudinally moving string and some problem in dynamics of winding sets [J]. Soviet Appl Mech,1983, 19(3):261-267.
[13]ULSOY A G. Vibration control in rotating or translating elastic systems [J]. Journal of Dynamic Systems, Measurement, and Control,1984,106(1):6-14.
[14]YANG B, MOTE C D JR. Active vibration control of the axially moving string in the S domain [J]. Journal of Applied Mechanics,1991,58(1):189-196.
[15]CHUNG C H, TAN C A. Active vibration control of the axially moving string by wave cancellation [J]. Journal of Vibration and Acoustic,1995,117(1):49-55.
[16]YING S, TAN C A. Active vibration control of the axially moving string using space feedforward and feedback controllers [J]. Journal of Vibration and Acoustic,1996,118(3):306-312.
[17]LEE S Y, MOTE C D JR. Vibration control of an axially moving string by boundary control [J]. Journal of Dynamic Systems, Measurement, and Control,1996,118(1):66-74.
[18]HUANG J S, WU J W, LU P Y. A study of moving string with partial state feedback [J]. International Journal of Mechanical Sciences,2002,44(9):1893-1905.
[19]HUANG J S, FUNG F R, CHEN D S. Application on variable structure control in the gyroscopic string vibration [J]. Trans Aeronaut Soc Rep China,1994,26(4):329-338.
[20]FUNG R F, LIAO C C. Application if variable structure control in the nonlinear string system[J]. International Journal of Mechanical Sciences,1995,37(9):985-993.
[21]FUNG R F, HUANG J S, WANG Y C, et al. Vibration reduction of the nonlinearly traveling string by a modified variable structure control with proportional and integral compensations [J]. International Journal of Mechanical Sciences,1998,40(6):493-506.
[22]FUNG R F, TSENG C C. Boundary control of an axially moving string via Lyapunov method[J]. Journal of Dynamic Systems, Measurement, and Control,1999,121(1):105-110.
[23]ZHU W D, NI J, HUANG J. Active control of translating media with arbitrarily varying length [J]. Journal of Vibration and Acoustic,2001,123:347-358.
[24]SHAHRUZ S M, KURMAJI D A. Vibration suppression of a non-linear axially moving string by boundary control [J]. Journal of Sound and Vibration,1997,201(1):145-152.
[25]SHAHRUZ S M. Boundary control of a nonlinear axially moving string [J]. International Journal of Nonlinear Control,2000,10(1):7-25.
[26]QUEIROZ M S, DAWSON D M, RAHN C D, et al. Adaptive vibration control of an axially moving string [J]. Journal of Vibration and Acoustic,1999,121(1):41-49.
[27]FUNG F R, WU J W, LU P Y. Adaptive boundary control of an axially moving string system [J]. Journal of Vibration and Acoustic,2002,124(1):193-198.
[28]HUANG J S, CHAO P C P, FUNG R F, et al. Parametric control of an axially moving string via fuzzy sliding-mode and fuzzy neural network methods [J]. Journal of Sound and Vibration,2003, 264:177-201.
[29]CHAO P C P, LAI C L. Boundary control of an axially moving string via fuzzy sliding-mode control and fuzzy neural network methods [J]. Journal of Sound and Vibration,2003,262: 795-813.
[30]JACK H. Theory of Functional Differential Equations [M]. New York:Spring-Verlag,1977.
[31]秦元勋,刘永清,王联等.带有时滞的动力系统的稳定性(第二版)[M].北京：科学出版社,1989.
[32]WIRKUS S, RAND R. The dynamics of two coupled van der Pol oscillators with delay coupling [J]. Nonlinear Dynamic,2002,30:205-221.
[33]RAGHOTHAMA A, NARAYANAN. Periodic response and chaos in nonlinear systems with parametric excitation and time delay [J]. Nonlinear Dynamic,2002,27:341-365.
[34]RAMANA REDDY D V, SEN A, JOHNSTON G L. Dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks [J]. Physica D,2000,144:335-357.
[35]CAMPBELL S A, BELAIR J, OHIRA T, et al. Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback [J]. Chaos,1995,5(4):640-645.
[36]XU J, CHUNG K W. Effects of time delayed position feedback on a Van der Pol-Duffing oscillator [J]. Physica D,2003,180:17-39.
[37]XU J, CHUNG K W, CHAN C L. An efficient method for studying weak resonant double Hopf bifurcation in nonlinear systems with delayed feedbacks [J]. SIAM Journal on Applied Dynamical Systems,2007,6(1):29-60.
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