轻质材料层合板的非线性动力学理论分析与实验研究
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摘要
随着航空航天飞行器的发展，飞行器独特的力学环境和性能要求对结构设计提出了新的课题，为适应这些要求，诸多轻质材料应运而生。轻质材料的研制已成为高超声速飞行器设计与制造的关键技术之一。高新技术的快速发展使人们已不再满足于材料单纯的轻质化，而是寻找兼有轻质化和其他某种或几种优良性能的先进材料以适应不同的需求。点阵材料被视为最具发展前景的轻质材料，其应用可以减轻飞行器的重量，保证结构的强度和刚度，同时还能够提供储油、散热和电子屏蔽等多种功能。
在飞行过程中，飞行器的薄壁构件会出现大振幅的非线性振动，这种振动会导致飞行器结构的严重损坏，从而导致灾难性的事故发生。当结构产生大幅振动时，仍然使用线性理论进行研究会对结果带来较大的误差，甚至得到完全错误的结果，所以使用非线性动力学方法研究点阵夹芯板等轻质材料结构的复杂振动特性，分析非线性振动现象的规律和机理具有十分重要的科学价值和实用价值。本文主要应用解析方法和数值方法研究了点阵夹芯板的非线性振动响应、非线性刚度特性等问题，分析了分叉和混沌运动，并且实验研究了碳纤维复合材料层合板的非线性动力学特性。论文的主要研究内容包括以下三个方面。
(1)研究了横向载荷和面内载荷联合作用下四边简支3D-Kagome点阵夹芯板的非线性动力学特性。利用拟夹层板分析方法将点阵夹芯板等效为一块由三层层片组成的复合层合板。基于von Karman板理论和Reddy高阶剪切变形理论推导出点阵夹芯板的非线性振动微分方程，利用Galerkin法得到了系统的两自由度非线性动力学方程，应用多尺度法获得了1:1内共振、1:2内共振和1:3内共振三种共振情况下的平均方程。基于所获得的两自由度非线性动力学方程和四维平均方程，通过数值方法研究了不同共振情况下点阵夹芯板的非线性振动响应，得到了系统的分叉图、波形图和相图。研究结果表明，随着激励幅值的增加，点阵夹芯板的运动状态可以呈现出周期运动和混沌运动之间交替变化的规律。
(2)根据平均方程获得了系统的频率响应函数，分析了1:2内共振情况下点阵夹芯板的非线性刚度特性和激励幅值作用下的振幅跳跃现象。研究结果表明，两阶模态之间的强耦合效应能够使模态原有的软(硬)刚度特性变为硬(软)刚度特性，并且能够影响振幅跳跃现象产生的频带。此外，还研究了其他两种共振情况下的幅-频响应特性。研究结果表明，三种共振情况下的幅-频响应差异较大。
(3)利用实验方法研究了四边简支碳纤维复合材料层合板的非线性振动特性。分析了复合材料层合板在定频步进激励幅值情况下振动状态的变化，分析了不同激励幅值下非线性振动现象以及层合板在整个过程中振幅的变化规律。此外，研究了复合材料层合板非线性刚度特性变化的机理，采用定幅缓慢扫频的激励方式，通过寻找在上下扫频过程中跳跃现象发生的频率，判断层合板的非线性刚度特性。对实验数据进行了频谱分析，研究结果表明，亚谐共振的发生应该是刚度特性变化的主要原因。
With the development of aerospace structures, the higher standards are proposed inthe structural design because of the special mechanical environment and performancerequirement of aircraft. In order to meet these requirements, many kinds of light-weightmaterials emerge as the times require. The development of the light-weight material turnsinto a key technology of design and manufacturing for hypersonic aircraft. Thelight-weight materials with several functions are paid more attention and the truss materialis viewed as the most promising light-weight material. The weight of the aircraft can belightened and the strength and stiffness of structures can be maintained by using the trussmaterial on the aircraft. In the mean time, this material can be used to provide a lot offunctions such as oil storage, cooling and electronic shielding.
The large amplitude nonlinear vibrations of plates often occur in the aircrafts,aero-turbines and aerospace vehicles, which will lead to the serious damages of structures.Because the linear theory is failed to analyze the large amplitude nonlinear vibrations ofstructures, research on nonlinear dynamic properties and the evolution law and mechanismof nonlinear phenomenon of light-weight sandwich plates have important scientific andpractical significance. In this paper, the nonlinear vibration response, hardening andsoftening types of nonlinearity of the truss core sandwich plate are investigated by usinganalytic and numerical methods. Furthermore, the nonlinear behaviors of the carbon fibersandwich plate are studied experimentally. The main contents of this dissertation are asfollows
(1) The nonlinear dynamics of simply supported at four edges3D-Kagome sandwichplate subjected to transverse and in-plane excitations are investigated. The truss coresandwich plate can be viewed as a continuum laminate according to the equivalentsandwich plate method. Based on the von Karman theory, Reddy third-order sheardeformation plate theory and the Galerkin technique, the two-degree-of-freedom ordinarydifferential equation of motion for the truss core sandwich plate is derived. Then, theaveraged equations of the system are obtained under three resonance case of1:1internalresonance,1:2internal resonance and1:3internal resonance by using the method ofmultiple scales. Numerical simulations are performed to study the vibrations of the trusscore sandwich plate. The results of numerical simulations exhibit the existence of theperiod, multi-period and chaotic responses with the variation of the excitations, whichdemonstrate that those motions appear alternately.
(2) The frequency-response functions are obtained based on the averaged equation inpolar form. The hardening and softening types of nonlinearity and the jump phenomenonwith the excitation amplitude are analyzed under1:2internal resonance. It is concluded that the hardening (softening) nonlinearity of certain mode can be change to softening(hardening) nonlinearity when the two modes are strongly coupled. It also can be seen thatthe frequency band at which the jump phenomenon with the excitation amplitude occurscan be changed by the coupling of two modes. In addition, the frequency-responsebehaviors are studied under1:1internal resonance and1:3internal resonance of the trusscore sandwich plate. It is seen that there are huge differences between thefrequence-response behaviors under the three resonance cases.
(3) The nonlinear dynamic behaviors of a simply supported carbon compositelaminated plate are investigated experimentally. The variation of vibration state is studiedunder the excitation of stepped amplitudes and fixed frequency in first part, the evolutionlaw of periodic and chaotic motions for the plate is obtained. The mechanism of thehardening and softening types of nonlinearity is analyzed in the second part, theexperimental frequency-response curves are plotted and the type of nonlinearity isobtained based on the test data of low speed frequency sweeps. The amplitude spectrumsof the test plate demonstrate that the change of the nonlinear trends may be caused by thesub-harmonic resonance.
在飞行过程中，飞行器的薄壁构件会出现大振幅的非线性振动，这种振动会导致飞行器结构的严重损坏，从而导致灾难性的事故发生。当结构产生大幅振动时，仍然使用线性理论进行研究会对结果带来较大的误差，甚至得到完全错误的结果，所以使用非线性动力学方法研究点阵夹芯板等轻质材料结构的复杂振动特性，分析非线性振动现象的规律和机理具有十分重要的科学价值和实用价值。本文主要应用解析方法和数值方法研究了点阵夹芯板的非线性振动响应、非线性刚度特性等问题，分析了分叉和混沌运动，并且实验研究了碳纤维复合材料层合板的非线性动力学特性。论文的主要研究内容包括以下三个方面。
(1)研究了横向载荷和面内载荷联合作用下四边简支3D-Kagome点阵夹芯板的非线性动力学特性。利用拟夹层板分析方法将点阵夹芯板等效为一块由三层层片组成的复合层合板。基于von Karman板理论和Reddy高阶剪切变形理论推导出点阵夹芯板的非线性振动微分方程，利用Galerkin法得到了系统的两自由度非线性动力学方程，应用多尺度法获得了1:1内共振、1:2内共振和1:3内共振三种共振情况下的平均方程。基于所获得的两自由度非线性动力学方程和四维平均方程，通过数值方法研究了不同共振情况下点阵夹芯板的非线性振动响应，得到了系统的分叉图、波形图和相图。研究结果表明，随着激励幅值的增加，点阵夹芯板的运动状态可以呈现出周期运动和混沌运动之间交替变化的规律。
(2)根据平均方程获得了系统的频率响应函数，分析了1:2内共振情况下点阵夹芯板的非线性刚度特性和激励幅值作用下的振幅跳跃现象。研究结果表明，两阶模态之间的强耦合效应能够使模态原有的软(硬)刚度特性变为硬(软)刚度特性，并且能够影响振幅跳跃现象产生的频带。此外，还研究了其他两种共振情况下的幅-频响应特性。研究结果表明，三种共振情况下的幅-频响应差异较大。
(3)利用实验方法研究了四边简支碳纤维复合材料层合板的非线性振动特性。分析了复合材料层合板在定频步进激励幅值情况下振动状态的变化，分析了不同激励幅值下非线性振动现象以及层合板在整个过程中振幅的变化规律。此外，研究了复合材料层合板非线性刚度特性变化的机理，采用定幅缓慢扫频的激励方式，通过寻找在上下扫频过程中跳跃现象发生的频率，判断层合板的非线性刚度特性。对实验数据进行了频谱分析，研究结果表明，亚谐共振的发生应该是刚度特性变化的主要原因。
With the development of aerospace structures, the higher standards are proposed inthe structural design because of the special mechanical environment and performancerequirement of aircraft. In order to meet these requirements, many kinds of light-weightmaterials emerge as the times require. The development of the light-weight material turnsinto a key technology of design and manufacturing for hypersonic aircraft. Thelight-weight materials with several functions are paid more attention and the truss materialis viewed as the most promising light-weight material. The weight of the aircraft can belightened and the strength and stiffness of structures can be maintained by using the trussmaterial on the aircraft. In the mean time, this material can be used to provide a lot offunctions such as oil storage, cooling and electronic shielding.
The large amplitude nonlinear vibrations of plates often occur in the aircrafts,aero-turbines and aerospace vehicles, which will lead to the serious damages of structures.Because the linear theory is failed to analyze the large amplitude nonlinear vibrations ofstructures, research on nonlinear dynamic properties and the evolution law and mechanismof nonlinear phenomenon of light-weight sandwich plates have important scientific andpractical significance. In this paper, the nonlinear vibration response, hardening andsoftening types of nonlinearity of the truss core sandwich plate are investigated by usinganalytic and numerical methods. Furthermore, the nonlinear behaviors of the carbon fibersandwich plate are studied experimentally. The main contents of this dissertation are asfollows
(1) The nonlinear dynamics of simply supported at four edges3D-Kagome sandwichplate subjected to transverse and in-plane excitations are investigated. The truss coresandwich plate can be viewed as a continuum laminate according to the equivalentsandwich plate method. Based on the von Karman theory, Reddy third-order sheardeformation plate theory and the Galerkin technique, the two-degree-of-freedom ordinarydifferential equation of motion for the truss core sandwich plate is derived. Then, theaveraged equations of the system are obtained under three resonance case of1:1internalresonance,1:2internal resonance and1:3internal resonance by using the method ofmultiple scales. Numerical simulations are performed to study the vibrations of the trusscore sandwich plate. The results of numerical simulations exhibit the existence of theperiod, multi-period and chaotic responses with the variation of the excitations, whichdemonstrate that those motions appear alternately.
(2) The frequency-response functions are obtained based on the averaged equation inpolar form. The hardening and softening types of nonlinearity and the jump phenomenonwith the excitation amplitude are analyzed under1:2internal resonance. It is concluded that the hardening (softening) nonlinearity of certain mode can be change to softening(hardening) nonlinearity when the two modes are strongly coupled. It also can be seen thatthe frequency band at which the jump phenomenon with the excitation amplitude occurscan be changed by the coupling of two modes. In addition, the frequency-responsebehaviors are studied under1:1internal resonance and1:3internal resonance of the trusscore sandwich plate. It is seen that there are huge differences between thefrequence-response behaviors under the three resonance cases.
(3) The nonlinear dynamic behaviors of a simply supported carbon compositelaminated plate are investigated experimentally. The variation of vibration state is studiedunder the excitation of stepped amplitudes and fixed frequency in first part, the evolutionlaw of periodic and chaotic motions for the plate is obtained. The mechanism of thehardening and softening types of nonlinearity is analyzed in the second part, theexperimental frequency-response curves are plotted and the type of nonlinearity isobtained based on the test data of low speed frequency sweeps. The amplitude spectrumsof the test plate demonstrate that the change of the nonlinear trends may be caused by thesub-harmonic resonance.
引文
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