大范围运动柔性梁非线性动力学
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摘要
本文系统地研究了大范围运动柔性梁的非线性动力学。涉及：大范围运动柔性梁的非线性动力学建模，轴向基础激励悬臂梁的周期振动，含内共振大范围直线运动梁的参激振动稳定性，参数激励与内激励联合作用的大范围直线运动梁的非线性动力行为，及窄带随机参数激励下直线运动梁的随机稳定性等问题。旨在全面地揭示所分析对象固有的非线性动力行为本质。具体内容如下：
第一章论述了大范围运动柔性体动力学及与其相关的各研究领域的研究现状，阐明了论文的立题目的及意义，并介绍了研究的主要内容与结构安排。
第二章基于Kane方程及假设模态，在考虑非线性广义作用力及非线性广义惯性力的基础上，导出了大范围运动柔性梁的非线性动力学控制方程(组)。该方程(组)中既有参数激励项，又有外激励项，既有二次非线性项，又存在着三次非线性项。在三次非线性项中，既有惯性项，又有几何项，且几何项除与梁的物理特性有关，还与梁的大范围转动紧密相连，为同类问题的相关研究提供了一个较完整的动力学模型。
第三章以基础激励悬臂梁的非线性动力行为的研究为切入点，与前期相关文献中有关建模理论的分析与比较，从一个侧面较充分地验证了本文有关分析对象的非线性动力学建模理论及结果的正确性。更为重要的是为后续研究工作的进一步深入开展提供了强有力的理论基础上的保障，同时，也完善了基础激励悬臂梁动力行为的研究。
第四章利用多尺度法并结合笛卡尔坐标变换，对基于Kane方程所建立起的包含有耦合三次几何及惯性非线性项的大范围直线运动梁动力学控制方程进行一次近似展开，着重对含内共振现象的大范围运动两端铰支梁参激振动平凡响应的稳定性进行了详尽的分析与研究，分析了分叉值处的 Hopf分叉类型，获得了一些新发现。
第五章同样利用多尺度法并结合笛卡尔坐标变换，对在一定直线运动加速度值下前两阶模态主参激共振或组合参激共振与一、二阶模态间3：1内共振联合作用的简支梁响应的定常解及其稳定性进行了较详尽的分析研究。同时，采用中心流形定理对调制方程进行了降维处理，分析了分叉值处的Hopf分叉类型，通过数值分析，揭示了研究对象所蕴涵的一系列内在的非线性动力学现象。
第六章仍以前几章所涉及的大范围直线运动梁为分析对象，在考虑了主参激共振、组合参激共振及内共振条件下，利用多尺度法，通过计算系统近似稳态响应的最大Lyapunov指数，对系统定常解，特别是平凡响应的随机稳定性问题，进
大范围运动柔性梁非线性动力学
行了一些探索性的研究，数值积分表明分析结果的有效性。
最后，在第七章中，作者对全文的主要工作、特点、创新与贡献之处进行了
概括与总结，并简列了本课题今后值得进一步研究的几个方向。
This dissertation systematically presents a study on the nonlinear dynamics of flexible beams undergoing a large overall motion, including the nonlinear dynamic modeling of these beams, the periodic vibration of a cantilever beam subject to axial movement of base, the dynamic stability analysis of parametrically excited slender beams undergoing a larger linear motion, the nonlinear dynamic behaviors of slender beams under the combination of both parametric excitation and internal excitation, and the largest Lyapunov exponent and the almost certain stability analysis of slender beams under the narrow-band random parametric excitation. The purpose of this dissertation is to gain an insight into the inherent nonlinear dynamics of flexible beams undergoing a large overall motion. The dissertation is organized as following.
Chapter 1 surveys the state-of-the-art of dynamics of flexible structures undergoing a large overall motion and the advance in the corresponding theories, and presents the significance, main contents and arrangement of the dissertation.
In Chapter 2, a set of nonlinear differential equations is established by using Kane's method for the flexible beams undergoing a large overall motion. Compared with the linear model where the generalized inertial force and generalized active force are linearized, the present model takes these nonlinear terms into consideration, which makes it possible for one to capture and understand their complicated nonlinear dynamics.
Chapter 3 focuses on the dynamic behaviors of a cantilever beam under an axial movement of its base. The results show that the nonlinear inertia terms produce a softening effect and play a significant role in the planar response of the second mode and the higher ones. On the other hand, the nonlinear geometric terms produce a hardening effect and dominate the planar response of the first mode. The validity of the present modeling results is clarified by comparing and calculating the corresponding coefficients in the present modeling method and other investigators' modeling methods, which, to a large extent, gives one the necessary theoretical security to continue the further studies.
In Chapter 4, the first order approximation to the solution of a set of nonlinear differential equations, which is established in Chapter 2 and governs the planar motion of flexible beams undergoing a large linear motion, are systematically derived via the method of multiple scales. In the case of a simply supported beam with 3:1 internal resonance between the first two modes, the dynamic stability of the trivial state of the system is investigated by using the Cartesian transformation in detail. The equations of approximate stability boundary are derived. Finally, the modulation equations are reduced to a two-dimensional system and the type of the Hopf bifurcations are determined in the corresponding vicinity of the bifurcations via the center manifold theorem and the limit cycles are found.
In Chapter 5, the nonlinear planar response of a simply supported flexible beam undergoing a large linear motion, which is seldom dealt with by other investigators, to a principal parametric resonance of either its first or second mode or a combination parametric resonance of the additive type of its first two modes is investigated. The comprehensive periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations and the center manifold theorem are also used to determine the type of the Hopf bifurcations. The numerously complicated nonlinear dynamic behaviors of the beam are revealed.
Chapter 6 presents the nonlinear dynamic behaviors of a simply supported flexible beam subject to narrow-band random parametric excitation, in which either the principal parametric resonance of its first mode or a combination parametric resonance of the additive type of its first two modes with or without 3:1 internal resonance between the first two modes is taken in to conside
第一章论述了大范围运动柔性体动力学及与其相关的各研究领域的研究现状，阐明了论文的立题目的及意义，并介绍了研究的主要内容与结构安排。
第二章基于Kane方程及假设模态，在考虑非线性广义作用力及非线性广义惯性力的基础上，导出了大范围运动柔性梁的非线性动力学控制方程(组)。该方程(组)中既有参数激励项，又有外激励项，既有二次非线性项，又存在着三次非线性项。在三次非线性项中，既有惯性项，又有几何项，且几何项除与梁的物理特性有关，还与梁的大范围转动紧密相连，为同类问题的相关研究提供了一个较完整的动力学模型。
第三章以基础激励悬臂梁的非线性动力行为的研究为切入点，与前期相关文献中有关建模理论的分析与比较，从一个侧面较充分地验证了本文有关分析对象的非线性动力学建模理论及结果的正确性。更为重要的是为后续研究工作的进一步深入开展提供了强有力的理论基础上的保障，同时，也完善了基础激励悬臂梁动力行为的研究。
第四章利用多尺度法并结合笛卡尔坐标变换，对基于Kane方程所建立起的包含有耦合三次几何及惯性非线性项的大范围直线运动梁动力学控制方程进行一次近似展开，着重对含内共振现象的大范围运动两端铰支梁参激振动平凡响应的稳定性进行了详尽的分析与研究，分析了分叉值处的 Hopf分叉类型，获得了一些新发现。
第五章同样利用多尺度法并结合笛卡尔坐标变换，对在一定直线运动加速度值下前两阶模态主参激共振或组合参激共振与一、二阶模态间3：1内共振联合作用的简支梁响应的定常解及其稳定性进行了较详尽的分析研究。同时，采用中心流形定理对调制方程进行了降维处理，分析了分叉值处的Hopf分叉类型，通过数值分析，揭示了研究对象所蕴涵的一系列内在的非线性动力学现象。
第六章仍以前几章所涉及的大范围直线运动梁为分析对象，在考虑了主参激共振、组合参激共振及内共振条件下，利用多尺度法，通过计算系统近似稳态响应的最大Lyapunov指数，对系统定常解，特别是平凡响应的随机稳定性问题，进
大范围运动柔性梁非线性动力学
行了一些探索性的研究，数值积分表明分析结果的有效性。
最后，在第七章中，作者对全文的主要工作、特点、创新与贡献之处进行了
概括与总结，并简列了本课题今后值得进一步研究的几个方向。
This dissertation systematically presents a study on the nonlinear dynamics of flexible beams undergoing a large overall motion, including the nonlinear dynamic modeling of these beams, the periodic vibration of a cantilever beam subject to axial movement of base, the dynamic stability analysis of parametrically excited slender beams undergoing a larger linear motion, the nonlinear dynamic behaviors of slender beams under the combination of both parametric excitation and internal excitation, and the largest Lyapunov exponent and the almost certain stability analysis of slender beams under the narrow-band random parametric excitation. The purpose of this dissertation is to gain an insight into the inherent nonlinear dynamics of flexible beams undergoing a large overall motion. The dissertation is organized as following.
Chapter 1 surveys the state-of-the-art of dynamics of flexible structures undergoing a large overall motion and the advance in the corresponding theories, and presents the significance, main contents and arrangement of the dissertation.
In Chapter 2, a set of nonlinear differential equations is established by using Kane's method for the flexible beams undergoing a large overall motion. Compared with the linear model where the generalized inertial force and generalized active force are linearized, the present model takes these nonlinear terms into consideration, which makes it possible for one to capture and understand their complicated nonlinear dynamics.
Chapter 3 focuses on the dynamic behaviors of a cantilever beam under an axial movement of its base. The results show that the nonlinear inertia terms produce a softening effect and play a significant role in the planar response of the second mode and the higher ones. On the other hand, the nonlinear geometric terms produce a hardening effect and dominate the planar response of the first mode. The validity of the present modeling results is clarified by comparing and calculating the corresponding coefficients in the present modeling method and other investigators' modeling methods, which, to a large extent, gives one the necessary theoretical security to continue the further studies.
In Chapter 4, the first order approximation to the solution of a set of nonlinear differential equations, which is established in Chapter 2 and governs the planar motion of flexible beams undergoing a large linear motion, are systematically derived via the method of multiple scales. In the case of a simply supported beam with 3:1 internal resonance between the first two modes, the dynamic stability of the trivial state of the system is investigated by using the Cartesian transformation in detail. The equations of approximate stability boundary are derived. Finally, the modulation equations are reduced to a two-dimensional system and the type of the Hopf bifurcations are determined in the corresponding vicinity of the bifurcations via the center manifold theorem and the limit cycles are found.
In Chapter 5, the nonlinear planar response of a simply supported flexible beam undergoing a large linear motion, which is seldom dealt with by other investigators, to a principal parametric resonance of either its first or second mode or a combination parametric resonance of the additive type of its first two modes is investigated. The comprehensive periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations and the center manifold theorem are also used to determine the type of the Hopf bifurcations. The numerously complicated nonlinear dynamic behaviors of the beam are revealed.
Chapter 6 presents the nonlinear dynamic behaviors of a simply supported flexible beam subject to narrow-band random parametric excitation, in which either the principal parametric resonance of its first mode or a combination parametric resonance of the additive type of its first two modes with or without 3:1 internal resonance between the first two modes is taken in to conside
引文
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81.Esmailzadeh E, Nakhaie-Jazar G Periodic behaviors of a cantilever beam with end mass subjected to harmonic base excitation. International Journal of Non-Linear Mechanics 1998, 33(4): 567-577
82.Esmailzadeh E, Jalili N. Parametric response of cantilever Timoshenko beams with tip mass under harmonic support motion. International Journal of Non-Linear Mechanics, 1998, 33(5): 765-781
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91.Knudsen J, Massih A R. Vibro-impact dynamics of a periodically forced beam. ASME Journal of Pressure Vessel Technology, 2000, 122(2): 210-221
92.Yigit A S, Christoforou A E Impact dynamics of composite beams. Composite Structures, 1995, 32(1-4): 187-195
93.Tseng C Y, Tung P C. Dynamics of a flexible beam with active nonlinear magnetic force. ASME Journal of Vibration and Acoustics, 1998, 120(1): 39-46
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80.Hyun S H, Yoo H H. Dynamic modeling and stability analysis of axially oscillating cantilever beams. Journal of Sound and Vibration, 1999, 228(3): 543-558
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91.Knudsen J, Massih A R. Vibro-impact dynamics of a periodically forced beam. ASME Journal of Pressure Vessel Technology, 2000, 122(2): 210-221
92.Yigit A S, Christoforou A E Impact dynamics of composite beams. Composite Structures, 1995, 32(1-4): 187-195
93.Tseng C Y, Tung P C. Dynamics of a flexible beam with active nonlinear magnetic force. ASME Journal of Vibration and Acoustics, 1998, 120(1): 39-46
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