梁结构非线性振动的最优化控制研究
摘要
本文提出一种非线性振动系统主共振和超谐共振的最优化控制方法，并将其应用于梁结构的非线性振动控制。提出非线性振动的压电自反馈最优化控制方法、压电时滞反馈最优化控制方法、磁时滞反馈最优化控制方法和分阶最优控制方法，研究了梁结构和参数激励系统的非线性振动控制。
论文的主要工作和创新性成果如下:
1.利用线性和非线性反馈控制器实现非线性系统的控制振动，提出一种确定非线性振动系统稳定振动的反馈增益取值范围的方法。利用多尺度法和平均法得到雅可比本征方程，进而得到非线性振动系统稳定条件，构造参数方程，根据参数方程有无实数解的条件得到反馈控制增益范围。将主共振和超谐共振有控制和无控制振动峰值的比值为减振的衰减率。以减振系统的衰减率和能量函数为目标函数，以稳定振动的反馈控制参数的范围为约束条件，利用最优化方法可计算得到衰减率和能量最小的速度、位移控制参数的最优取值。设计线性和非线性最优化控制器控制非线性振动系统的动力学行为。提出一种梁结构非线性振动的压电自反馈的最优化控制方法。利用压电材料压电效应和逆压电效应来实现结构的自反馈闭环控制。由悬臂梁主共振和超谐共振实验实测数据计算出最优化控制参数，设计非线性主共振振动实验控制系统，进行减振控制实验研究。
2.提出一种结构非线性振动的时滞自反馈最优化控制方法。分别以压电元件为感受器和驱动器，由受控系统和时滞控制系统组成闭合回路，应用时滞自反馈最优化控制方法控制梁的非线性振动。
3.研究了参数激励系统的时滞反馈最优化控制。依据平均法得到稳态响应振幅和相位的平均方程。以非线性振动能量比值的衰减率为优化目标函数，以非线性振动系统稳定振动条件、最值条件和最优时滞条件为约束条件，利用最优化方法计算得到最佳线性和非线性反馈控制增益参数。
4.提出非线性方程的分阶最优控制策略，并将其应用于悬臂梁非线性振动的压电减振控制，通过空间解耦，得到状态空间方程，设计非线性分阶控制器，对该减振系统进行分阶最优控制。
5.研究纳米谐振梁非线性特性产生的物理机制，分析非局部效应对固有频率的影响，探讨非局部效应对主共振幅频特性曲线的影响，对纳米谐振梁的振动特性进行了分析。以压电结构为控制信号发生和驱动元件，研究考虑纳米尺度非局部效应的纳米梁的振动控制，研究非局部效应参数与控制参数和时滞的影响关系。提出一种非接触的碳纳米管的主共振响应的磁时滞最优化控制方法。利用发生褶皱变形碳纳米管弯曲力矩的二阶导数和梁的弯曲理论建立碳纳米管梁的微分方程。设计最优控制器控制碳纳米管系统的非线性振动。
An optimal control method is presented for the vibration control of the primary and super-harmonic resonances of nonlinear vibration system. The nonlinear vibration of beams is controlled bythe control methods. Some new control methods, such as optimal piezoelectric self-feedback, optimalpiezoelectric delayed self-feedback, optimal magnetic delayed feedback and non-linear grade optimalcontrol scheme, are proposed in the control of the nonlinear vibration. The nonlinear vibrations offlexible structures and nano structures are mitigated using the control methods.
The main works and novel researched performed in this dissertation include:
1. The optimal control method is utilized to control the vibration of the nonlinear system withlinear and nonlinear feedback controllers. The stable conditions of the nonlinear vibration systems aregiven from Jacobi eigenvalue equations. The equations of parameters function are constructed. Therange of feedback gains for a stable vibration is gotten according to the conditions of solutions. Theattenuation of vibration reduction is defined by the ratio of peak value for the primary resonance andharmonic resonance with control and without control. Taking the attenuation of vibration reductionsystem and the energy function as objective functions and taking the stable feedback controlparameters as constraint conditions, the optimal control parameters of the velocity and displacementare calculated by the optimization method. The linear and nonlinear optimal controllers are designedto control the dynamic behavior of the nonlinear vibration system. An optimal self-feedback controlmethodology is provided to mitigate the primary and superharmonic resonances of a cantilever beam.The controller is designed to control the vibration of primary and superharmonic resonances ofintelligent structures. An experimental control system is designed and control experiment is carriedout.
2. The primary and superharmonic resonances of flexible beams are mitigated by a piezoelectricoptimal time delay self-feedack control methodology. The piezoelectric optimal controllers aredesigned to control the dynamic behaviors of the nonlinear dynamical systems. It is found that boththe optimal feedback gains and time delay can change the control performance.
3. The optimal delayed feedback control of the nonlinear vibration is studied with parametricexcitation. The equations of the amplitude and phase are obtained by the average method. A vibrationdecay rate is defined by the energy ratio of the nonlinear vibration with control and without control.Taking the decay rate as the objective function and the stable conditions and the optimal delayconditions as constraints, the linear and non-linear optimal feedback control parameters arecalculated by optimization method.
4. A non-linear grade optimal control scheme is proposed and used in the piezoelectric vibrationreduction control of non-linear cantilever beam. The differential equation is linearized into a set oflinear equations using perturbation method. The state space equations are obtained by decoupling inthe space coordinates. The non-linear grade controllers are designed to control the nonlinear vibration.
5. The natural frequency is gotten and the principal resonance is studied considering non-localeffect and axial nonlinear elongation. The numerical results show that the nonlocal effect has aneffect on the natural frequency and the relationship between frequency and amplitude. Using thepiezoelectric elements to sign and control driving actuators, the influence of the nonlocal effects ofparameters and time delay on the control parameters is studied. An optimal maganetic delayed self-feedback control method is provided to mitigate the primary resonance of a single-walled carbonnanotube. The nonlinear governing equations of motion for the SWCNT under longitudinal magneticfield are derived. The optimal controllers are designed to control the dynamic behaviors of thenonlinear vibration systems.
论文的主要工作和创新性成果如下:
1.利用线性和非线性反馈控制器实现非线性系统的控制振动，提出一种确定非线性振动系统稳定振动的反馈增益取值范围的方法。利用多尺度法和平均法得到雅可比本征方程，进而得到非线性振动系统稳定条件，构造参数方程，根据参数方程有无实数解的条件得到反馈控制增益范围。将主共振和超谐共振有控制和无控制振动峰值的比值为减振的衰减率。以减振系统的衰减率和能量函数为目标函数，以稳定振动的反馈控制参数的范围为约束条件，利用最优化方法可计算得到衰减率和能量最小的速度、位移控制参数的最优取值。设计线性和非线性最优化控制器控制非线性振动系统的动力学行为。提出一种梁结构非线性振动的压电自反馈的最优化控制方法。利用压电材料压电效应和逆压电效应来实现结构的自反馈闭环控制。由悬臂梁主共振和超谐共振实验实测数据计算出最优化控制参数，设计非线性主共振振动实验控制系统，进行减振控制实验研究。
2.提出一种结构非线性振动的时滞自反馈最优化控制方法。分别以压电元件为感受器和驱动器，由受控系统和时滞控制系统组成闭合回路，应用时滞自反馈最优化控制方法控制梁的非线性振动。
3.研究了参数激励系统的时滞反馈最优化控制。依据平均法得到稳态响应振幅和相位的平均方程。以非线性振动能量比值的衰减率为优化目标函数，以非线性振动系统稳定振动条件、最值条件和最优时滞条件为约束条件，利用最优化方法计算得到最佳线性和非线性反馈控制增益参数。
4.提出非线性方程的分阶最优控制策略，并将其应用于悬臂梁非线性振动的压电减振控制，通过空间解耦，得到状态空间方程，设计非线性分阶控制器，对该减振系统进行分阶最优控制。
5.研究纳米谐振梁非线性特性产生的物理机制，分析非局部效应对固有频率的影响，探讨非局部效应对主共振幅频特性曲线的影响，对纳米谐振梁的振动特性进行了分析。以压电结构为控制信号发生和驱动元件，研究考虑纳米尺度非局部效应的纳米梁的振动控制，研究非局部效应参数与控制参数和时滞的影响关系。提出一种非接触的碳纳米管的主共振响应的磁时滞最优化控制方法。利用发生褶皱变形碳纳米管弯曲力矩的二阶导数和梁的弯曲理论建立碳纳米管梁的微分方程。设计最优控制器控制碳纳米管系统的非线性振动。
An optimal control method is presented for the vibration control of the primary and super-harmonic resonances of nonlinear vibration system. The nonlinear vibration of beams is controlled bythe control methods. Some new control methods, such as optimal piezoelectric self-feedback, optimalpiezoelectric delayed self-feedback, optimal magnetic delayed feedback and non-linear grade optimalcontrol scheme, are proposed in the control of the nonlinear vibration. The nonlinear vibrations offlexible structures and nano structures are mitigated using the control methods.
The main works and novel researched performed in this dissertation include:
1. The optimal control method is utilized to control the vibration of the nonlinear system withlinear and nonlinear feedback controllers. The stable conditions of the nonlinear vibration systems aregiven from Jacobi eigenvalue equations. The equations of parameters function are constructed. Therange of feedback gains for a stable vibration is gotten according to the conditions of solutions. Theattenuation of vibration reduction is defined by the ratio of peak value for the primary resonance andharmonic resonance with control and without control. Taking the attenuation of vibration reductionsystem and the energy function as objective functions and taking the stable feedback controlparameters as constraint conditions, the optimal control parameters of the velocity and displacementare calculated by the optimization method. The linear and nonlinear optimal controllers are designedto control the dynamic behavior of the nonlinear vibration system. An optimal self-feedback controlmethodology is provided to mitigate the primary and superharmonic resonances of a cantilever beam.The controller is designed to control the vibration of primary and superharmonic resonances ofintelligent structures. An experimental control system is designed and control experiment is carriedout.
2. The primary and superharmonic resonances of flexible beams are mitigated by a piezoelectricoptimal time delay self-feedack control methodology. The piezoelectric optimal controllers aredesigned to control the dynamic behaviors of the nonlinear dynamical systems. It is found that boththe optimal feedback gains and time delay can change the control performance.
3. The optimal delayed feedback control of the nonlinear vibration is studied with parametricexcitation. The equations of the amplitude and phase are obtained by the average method. A vibrationdecay rate is defined by the energy ratio of the nonlinear vibration with control and without control.Taking the decay rate as the objective function and the stable conditions and the optimal delayconditions as constraints, the linear and non-linear optimal feedback control parameters arecalculated by optimization method.
4. A non-linear grade optimal control scheme is proposed and used in the piezoelectric vibrationreduction control of non-linear cantilever beam. The differential equation is linearized into a set oflinear equations using perturbation method. The state space equations are obtained by decoupling inthe space coordinates. The non-linear grade controllers are designed to control the nonlinear vibration.
5. The natural frequency is gotten and the principal resonance is studied considering non-localeffect and axial nonlinear elongation. The numerical results show that the nonlocal effect has aneffect on the natural frequency and the relationship between frequency and amplitude. Using thepiezoelectric elements to sign and control driving actuators, the influence of the nonlocal effects ofparameters and time delay on the control parameters is studied. An optimal maganetic delayed self-feedback control method is provided to mitigate the primary resonance of a single-walled carbonnanotube. The nonlinear governing equations of motion for the SWCNT under longitudinal magneticfield are derived. The optimal controllers are designed to control the dynamic behaviors of thenonlinear vibration systems.
引文
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