工程柔性结构振动控制中的时滞动力学研究
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摘要
在土木、海洋和航天工程领域,为了满足实际工程需要,采用了不同的控制措施对柔性结构的振动进行控制。然而,控制器本身、信号采集处理、控制律运算和信号传输到作动器响应等都存在一定的时滞,尽管实际控制中时滞很小,但仍会使作动器在受控系统不需能量时向其输入,从而导致控制性能降低,结构失稳。因此,研究柔性结构控制系统中的时滞动力学对控制系统的优化设计具有明显的意义。此外,柔性结构振动控制中的时滞问题已经成为受控结构动力响应及其相关控制设计的重要组成部分。
为了全面揭示时滞因素在柔性结构振动控制系统中的作用机制,本论文以工程中具有代表性的柔性结构(索、梁及索-梁组合结构)为对象开展研究。基于Hamilton原理,针对索、梁及索-梁组合结构受控系统建立连续模型,同时考虑MR阻尼器-结构振动控制系统中的时滞因素,以及采用时滞反馈控制策略,引入时滞微分方程的基本理论,结合定量分析方法(多尺度法)和定性分析方法(中心流形法和范式理论),对受控系统进行深入研究。通过研究,全面揭示工程柔性结构振动控制系统中的时滞效应,并在理论分析与数值计算的基础上,分析了受控系统各参数(如时滞,控制增益等)对控制效果的影响,为控制系统的优化设计提供理论基础。本论文的主要研究工作如下：
1.在广泛查阅国内外文献的基础上,对工程柔性结构中具有代表性的索,梁及索-梁组合结构的非线性动力学及其振动控制研究进行了综述,并着重突出了控制系统中的时滞问题。
2.为研究MR阻尼器-斜拉索振动控制系统的时滞效应,探究其对结构非线性响应的影响,本论文基于Hamilton原理建立了考虑时滞作用下的MR阻尼器-斜拉索振动控制系统连续模型,利用多尺度法,对受控系统的面内面外1：1内共振及其分岔进行了研究,并采用打靶法、拟弧长延拓法等方法进行了数值分析。
3.通过选取更为精确描述安装MR阻尼器后斜拉索的振型函数,建立了MR阻尼器-斜拉索受控系统模型和运动方程。运用Galerkin方法将偏泛函微分动力系统离散为时滞微分动力系统,并结合中心流形法和范式理论,从而实现了频率不等的高维非线性系统从无穷维空间到低维空间的转换,根据降维后的振幅截断方程,讨论了原系统的非线性动力学行为,进而通过平均法处理,得到了外部荷载作用下的平均化方程。
4.基于加速度时滞反馈控制策略及Euler-Bernoulli梁理论,建立了轴力作用下弹性支座压电耦合梁的非线性动力学模型。通过模态分析和线性稳定性分析,得到了压电耦合作用时滞反馈条件下的系统稳定性条件。采用Galerkin方法和多尺度法,从理论上得到了时滞动力系统的分岔响应。
5.基于Hamilton原理,利用动静法精细分析了索-梁的连接条件和边界条件,建立了索-梁组合结构面内运动方程,进而利用时滞反馈控制对索-梁组合结构的大幅振动进行控制。着重探讨时滞及控制增益对受控系统的影响,从而为索-梁组合结构的振动控制提供一种新思路。
上述研究成果表明：时滞是振动控制系统中的不容忽视的一个重要因素；系统在小时滞作用下具有鲁棒性,而随着时滞量的增大,系统可能呈现出复杂的非线性动力学行为(如拟周期运动、Hopf-Hopf分岔和混沌等),甚至导致结构失稳；另一方面,可以考虑通过调整控制系统中的参数来避免失稳及复杂运动的产生,达到改善控制性能的目的。本文的理论研究成果可作为振动控制设计理论在实际工程应用中的理论依据,所建议的设计方法也可作为提高控制系统性能有益补充和参考。
In order to satisfy the practical engineering requirement, diferent control measures invibration control of the flexible structure have been adopted in the fields of civil engineering,ocean engineering and aerospace engineering. However, a fixed time delay existing due tothe online data acquisition, filtering, manipulation of the digital data inside the controlprocessor, calculation of the required control action and the signal transmission from thecomputer to the actuator. Small as the time delay is in reality control system, but it maycause a detrimental efect on the stability of the controlled structure. So time delay dynamicsis very important to optimize the design of the control system. Moreover, the time delayproblem is an important part of the dynamic response and design of controlled structures.
To make a further address the mechanisms of the time delay in the controlled flex-ible structure, representative flexible engineering structure (cable, beam and cable-beamstructure) are investigated. Based on the Hamilton principle, the continuous dynamicalmodel of cable, beam and cable-beam structure are derived by considering the time delayfactors in the MR damper-stay cable control system, and by using delayed feedback con-trol strategy, and introducing the basic theory of delay diferential equation (DDE). Thenthe control system are investigated by combined the quantitative analysis method (multiplescales method) and qualitative analysis method (center manifold method and normal formtheory). We study the efectiveness of the time delay in the control system, and based onthe theoretical analysis and numerical computing, and analyze the influence of parameters(such as time delay, control gain) on the control efect, and provide a theoretical bases forthe optimized design of the control system. The contents of this thesis are as follows:
1. Based on the extensively reviewing literatures, the nonlinear dynamics and it’s vibra-tion control studies of cable, beam and cable-beam structure (the representative of theflexible engineering structure) are stated, and the delay problem in the control systemare highlighted, respectively.
2. In order to study the efectiveness of the time delay in the MR damper-stay cablecontrol system and explore the nonlinear response of the structure, based on theHamilton principle, a dynamical model of the MR damper-stay cable system is derivedand the Galerkin method is used to obtain the time delay dynamics system. By usingthe multiple scales method, the in-plane and out-of-plane1:1internal resonance andthe bifurcation of the control system are studied. Moreover, the numerical results areobtained by using the shooting method and arc-length continuation method.
3. By choosing a more precise description of the cables vibration mode function in theMR dampers, the MR damper-cable stayed control system model and the equationsof motion are derived and the Galerkin method is used to obtain the time delaydynamics system, and the center manifold method and normal form theory are used toa high dimensional nonlinear systems with unequal frequency conversion from infinitedimensional space to low dimensional space. Then, based on amplitude truncatedequation after dimensionality reduction, nonlinear dynamical behavior of the originalsystem are studied, and the averaging equations under external loads are obtained bysing the average method.
4. Based on acceleration delayed feedback control strategy and Euler-Bernoulli beamtheory, the nonlinear dynamic model of the elastic support axial force piezoelectriccoupling beam are obtained. Via conducing the modal analysis and linear stabilityanalysis, the stability conditions of system under piezoelectric coupling and delayedfeedback are obtained. The Galerkin method and multiple scale method are used tostudy the bifurcation response of the time delay system.
5. Based on the Hamilton principle, the connection and boundary conditions of the cable-beam structure are investigated by using the statics and dynamics method, and thein-plane motion equations are derived. Moreover, the large amplitude vibration ofcable-beam structure are controlled by using the delayed feedback control. The studyprovide an idea for the vibration control of the cable-beam structure by focused onthe influence of the time delay in the controlled system.
The results show that: the time delay is an important unignorable factor in the vibrationcontrol system with robust under the small time delay efect. As the time delay increas-ing, the system may exhibit complex nonlinear dynamics behavior (such as quasi-periodicmotion, Hopf-Hopf bifurcation and chaos, etc) or even lead to instability of structural. Byadjusting the control system parameters, instability and complex movement of the systemcan be avoided, and the control performance can be improved. This study can be theoreticalbasis in practical engineering applications, and the design method is also be recommendedas a useful supplement to improve the performance of the control system.
为了全面揭示时滞因素在柔性结构振动控制系统中的作用机制,本论文以工程中具有代表性的柔性结构(索、梁及索-梁组合结构)为对象开展研究。基于Hamilton原理,针对索、梁及索-梁组合结构受控系统建立连续模型,同时考虑MR阻尼器-结构振动控制系统中的时滞因素,以及采用时滞反馈控制策略,引入时滞微分方程的基本理论,结合定量分析方法(多尺度法)和定性分析方法(中心流形法和范式理论),对受控系统进行深入研究。通过研究,全面揭示工程柔性结构振动控制系统中的时滞效应,并在理论分析与数值计算的基础上,分析了受控系统各参数(如时滞,控制增益等)对控制效果的影响,为控制系统的优化设计提供理论基础。本论文的主要研究工作如下：
1.在广泛查阅国内外文献的基础上,对工程柔性结构中具有代表性的索,梁及索-梁组合结构的非线性动力学及其振动控制研究进行了综述,并着重突出了控制系统中的时滞问题。
2.为研究MR阻尼器-斜拉索振动控制系统的时滞效应,探究其对结构非线性响应的影响,本论文基于Hamilton原理建立了考虑时滞作用下的MR阻尼器-斜拉索振动控制系统连续模型,利用多尺度法,对受控系统的面内面外1：1内共振及其分岔进行了研究,并采用打靶法、拟弧长延拓法等方法进行了数值分析。
3.通过选取更为精确描述安装MR阻尼器后斜拉索的振型函数,建立了MR阻尼器-斜拉索受控系统模型和运动方程。运用Galerkin方法将偏泛函微分动力系统离散为时滞微分动力系统,并结合中心流形法和范式理论,从而实现了频率不等的高维非线性系统从无穷维空间到低维空间的转换,根据降维后的振幅截断方程,讨论了原系统的非线性动力学行为,进而通过平均法处理,得到了外部荷载作用下的平均化方程。
4.基于加速度时滞反馈控制策略及Euler-Bernoulli梁理论,建立了轴力作用下弹性支座压电耦合梁的非线性动力学模型。通过模态分析和线性稳定性分析,得到了压电耦合作用时滞反馈条件下的系统稳定性条件。采用Galerkin方法和多尺度法,从理论上得到了时滞动力系统的分岔响应。
5.基于Hamilton原理,利用动静法精细分析了索-梁的连接条件和边界条件,建立了索-梁组合结构面内运动方程,进而利用时滞反馈控制对索-梁组合结构的大幅振动进行控制。着重探讨时滞及控制增益对受控系统的影响,从而为索-梁组合结构的振动控制提供一种新思路。
上述研究成果表明：时滞是振动控制系统中的不容忽视的一个重要因素；系统在小时滞作用下具有鲁棒性,而随着时滞量的增大,系统可能呈现出复杂的非线性动力学行为(如拟周期运动、Hopf-Hopf分岔和混沌等),甚至导致结构失稳；另一方面,可以考虑通过调整控制系统中的参数来避免失稳及复杂运动的产生,达到改善控制性能的目的。本文的理论研究成果可作为振动控制设计理论在实际工程应用中的理论依据,所建议的设计方法也可作为提高控制系统性能有益补充和参考。
In order to satisfy the practical engineering requirement, diferent control measures invibration control of the flexible structure have been adopted in the fields of civil engineering,ocean engineering and aerospace engineering. However, a fixed time delay existing due tothe online data acquisition, filtering, manipulation of the digital data inside the controlprocessor, calculation of the required control action and the signal transmission from thecomputer to the actuator. Small as the time delay is in reality control system, but it maycause a detrimental efect on the stability of the controlled structure. So time delay dynamicsis very important to optimize the design of the control system. Moreover, the time delayproblem is an important part of the dynamic response and design of controlled structures.
To make a further address the mechanisms of the time delay in the controlled flex-ible structure, representative flexible engineering structure (cable, beam and cable-beamstructure) are investigated. Based on the Hamilton principle, the continuous dynamicalmodel of cable, beam and cable-beam structure are derived by considering the time delayfactors in the MR damper-stay cable control system, and by using delayed feedback con-trol strategy, and introducing the basic theory of delay diferential equation (DDE). Thenthe control system are investigated by combined the quantitative analysis method (multiplescales method) and qualitative analysis method (center manifold method and normal formtheory). We study the efectiveness of the time delay in the control system, and based onthe theoretical analysis and numerical computing, and analyze the influence of parameters(such as time delay, control gain) on the control efect, and provide a theoretical bases forthe optimized design of the control system. The contents of this thesis are as follows:
1. Based on the extensively reviewing literatures, the nonlinear dynamics and it’s vibra-tion control studies of cable, beam and cable-beam structure (the representative of theflexible engineering structure) are stated, and the delay problem in the control systemare highlighted, respectively.
2. In order to study the efectiveness of the time delay in the MR damper-stay cablecontrol system and explore the nonlinear response of the structure, based on theHamilton principle, a dynamical model of the MR damper-stay cable system is derivedand the Galerkin method is used to obtain the time delay dynamics system. By usingthe multiple scales method, the in-plane and out-of-plane1:1internal resonance andthe bifurcation of the control system are studied. Moreover, the numerical results areobtained by using the shooting method and arc-length continuation method.
3. By choosing a more precise description of the cables vibration mode function in theMR dampers, the MR damper-cable stayed control system model and the equationsof motion are derived and the Galerkin method is used to obtain the time delaydynamics system, and the center manifold method and normal form theory are used toa high dimensional nonlinear systems with unequal frequency conversion from infinitedimensional space to low dimensional space. Then, based on amplitude truncatedequation after dimensionality reduction, nonlinear dynamical behavior of the originalsystem are studied, and the averaging equations under external loads are obtained bysing the average method.
4. Based on acceleration delayed feedback control strategy and Euler-Bernoulli beamtheory, the nonlinear dynamic model of the elastic support axial force piezoelectriccoupling beam are obtained. Via conducing the modal analysis and linear stabilityanalysis, the stability conditions of system under piezoelectric coupling and delayedfeedback are obtained. The Galerkin method and multiple scale method are used tostudy the bifurcation response of the time delay system.
5. Based on the Hamilton principle, the connection and boundary conditions of the cable-beam structure are investigated by using the statics and dynamics method, and thein-plane motion equations are derived. Moreover, the large amplitude vibration ofcable-beam structure are controlled by using the delayed feedback control. The studyprovide an idea for the vibration control of the cable-beam structure by focused onthe influence of the time delay in the controlled system.
The results show that: the time delay is an important unignorable factor in the vibrationcontrol system with robust under the small time delay efect. As the time delay increas-ing, the system may exhibit complex nonlinear dynamics behavior (such as quasi-periodicmotion, Hopf-Hopf bifurcation and chaos, etc) or even lead to instability of structural. Byadjusting the control system parameters, instability and complex movement of the systemcan be avoided, and the control performance can be improved. This study can be theoreticalbasis in practical engineering applications, and the design method is also be recommendedas a useful supplement to improve the performance of the control system.
引文
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