【作者】 刘梅；

【导师】 曹登庆；

【作者基本信息】 哈尔滨工业大学 ， 力学， 2021， 博士

【摘要】 随着我国航天重大工程的逐步实施,航天器结构正朝着大型化、轻量化和多功能化等方向发展,桁架式空间可展结构因具有质量轻、刚度-质量比大、收纳率高等优点而被广泛应用于航天等工程领域。桁架结构具有自由度高、柔性大、构型复杂等特点,为满足大型空间结构动力学分析的高精度、高可靠性要求,其动力学建模、非线性振动与振动控制等问题越来越复杂且难于处理。因此,发展大型空间可展桁架结构的等效连续体建模方法,确保等效模型的准确性、有效性及高效性,在等效建模的基础上开展线性与非线性振动及其控制的研究,对于大型桁架结构的非线性动力学机理分析和低阶控制律设计具有重要的理论指导意义和工程实用价值。本文基于能量等效原理,建立了类梁桁架结构的等效动力学模型,以解析方法和数值仿真为手段,获取了等效梁模型的固有频率和耦合振动模态;考虑桁架结构的几何非线性特性,建立了等效非线性梁动力学模型,分别研究了非线性桁架结构和等效非线性梁模型在不同形式激励下的动力学特性。与原桁架结构有限元仿真结果的分析和对比表明,这里所提出的等效模型具有求解精度高和速度快等优点。采用线性二次型调节器(LQR)最优控制方法,设计了等效梁模型的振动控制律,基于设计的控制律实现了类梁桁架系统的振动抑制。具体研究内容如下:针对具有初应力的刚性连接周期性三棱柱类梁桁架结构,依据动力学假设和在周期单元中心处Taylor展开方法,推导空间周期单元的位移表达式。基于能量等效原理和静力凝聚法建立三维空间桁架的等效各向异性Timoshenko梁模型,获取等效梁模型的刚度和质量矩阵,进一步求得等效梁模型的固有频率和模态振型,并与文献及原桁架结构的有限元仿真结果分析和对比,验证了所建立等效梁模型的精确性和有效性,讨论了初应力和尺寸参数对等效梁模型精度的影响。与传统有限元方法相比,等效模型的建立更便于桁架结构的动力学特性分析与控制器设计,并且能够极大地节省计算时间。桁架结构的非对称性可导致系统存在耦合振动行为。基于这里提出的等效建模方法,采用Hamilton原理推导了等效梁模型的两组运动控制微分方程,分别呈现出弯曲-扭转和弯曲-拉伸耦合振动现象。采用精确解析法求解了两组耦合振动PDEs,获取了自由-自由和固定-自由边界条件下等效梁模型的固有频率和模态振型,并与原桁架结构有限元仿真结果分析和对比,验证了所建立等效梁模型和解析求解方法的精确性,并说明了耦合振动效应的强弱。针对大型细长柔性桁架结构,研究表明其几何非线性影响因素不可忽略,采用传统有限元方法计算桁架的非线性动力学响应耗时,且不便于系统非线性动力学机理的解析分析。考虑桁架结构的几何非线性,在等效梁模型基础上引入von Karman非线性应变-位移关系,建立了两端铰支非线性桁架的等效非线性梁模型,利用Hamilton原理和Galerkin方法获取等效梁模型的低阶离散动力学模型。为验证这里提出的等效非线性梁模型的有效性,从静力学和动力学角度分析和对比了不同载荷工况下,原桁架结构和等效非线性梁模型的非线性动态特性。对比求解非线性动力学响应的计算时间表明,等效非线性梁模型比传统有限元模型更具高效性。根据等效非线性梁模型的频率-响应曲线特性,分析了模态阶数对系统动力学响应的影响。采用高自由度的大型桁架有限元模型不便于系统的振动控制器设计,利用模态振型方法能够极大地降低系统的自由度,从而方便地设计系统的低阶控制律。基于这里提出的等效动力学建模方法,建立了悬臂类梁桁架结构的等效梁模型并采用解析法求解了其模态振型的显示表达式,获得等效梁模型的低阶离散动力学模型,针对周期激励、脉冲激励等外部载荷对结构造成的结构振动,根据等效梁模型设计了LQR振动控制器,同时将该控制律通过ANSYS有限元仿真软件施加到原类梁桁架系统。数值算例的对比和分析表明,基于等效梁模型设计的振动控制器能够有效地抑制原类梁桁架结构的振动位移。更多还原

【Abstract】 With the gradual implementation of the major projects in aerospace engineering of our country,spacecraft structure is heading towards the direction of large-scale,lightweight and multi-function,and thus large flexible deployable truss structures are broadly used in space applications due to their outstanding properties of light weight,large stiffness-to-mass ratio,improved vibration resistance,and high packaging efficiency.Truss structures have the features of high degree of freedom,large flexibility and complex configuration.To satisfy the high precision and high reliability demands of the dynamic analysis of the large space structure,the studies on dynamic modeling,nonlinear vibration and vibration control are getting more and more complicated and difficult to deal with.Therefore,it is necessary to develop the equivalent continuum modeling method of the large space deployable structure to guarantee the accuracy,validity and high-efficiency,as well as study the linear and nonlinear dynamic response and vibration control based on the equivalent continuum modeling method.On this basis,it has important theoretical guiding significance and engineering practical values to do research on analysis the nonlinear mechanism of the large trusses in analytical method and design the low-order control law conveniently.In this thesis,the equivalent dynamic models of the beamlike truss structures are established.Then natural frequencies and mode shapes are obtained employing the analytical derivation and numerical simulation.On this basis,considering the geometric nonlinearity of the tuss strucure,an equivalent nonlinear beam model is proposed and the dynamic characteristics for nonlinear truss system and equivalent nonlinear beam model are analyzed under different exciations.The comparison of finite element simulation results for original truss strucutre demonstrates that the proposed equivalent models have the characteristics of high precision and fast speed in solution process.The vibration controller of the equivalent beam model is designed utilizing the Linear Quadratic Regulator(LQR)control method.The vibration suppression of the beamlike truss system also be implemented using the control law designed by the equivalent beam model.The detailed research content is as follows:The displacement expressions of the spatial repeating element for rigid-jointed triangular prism truss with initial stress are derived using the kinematic assumptions and the Taylor series expansion at the center of the spatial repeating element.The equivalent anisotropic Timoshenko beam model of 3-Dimensional truss is proposed according to the energy equivalence principle and the static condensation method.Then,the stiffness and mass matrices of the equivalent beam model are obtained to achieve its natural frequencies and mode shapes.Comparing the results with those of finite element simulation for original truss structure,the accuracy and validity of the proposed equivalent beam model are checked.Also,the influence of the initial stress and size parameter on the precision of the equivalent beam model is discussed.The establishment of the equivalent dynamic models are more convenient for dynamic characteristic analysis and controler design,as well as can reduce significantly computational time,which are the advantages that the traditional finite element method do not have.There exist coupled vibration phenomenon caused by the asymmetry of the truss structure.Based on the proposed equivalent modeling method,the Hamilton’s principle is carried out to formulate the governing partial differential equations of motion for the equivalent beam model which is divided into two sets of PDEs describing bending-torsion and bending-extension couplings respectively.An exact analytical method is developed in this paper to solve the two sets of coupled motion equations to obtain the natural frequencies and mode shapes of the cantilevered and free-free equivalent beam model.The accuracy of the equivalent beam model and the analytical method is verified by comparing with the finite element simulation results of the original truss structure,and the coupled vibration effect is analyzed.For the large flexible beamlike truss structures,the nonlinear effect cannot be ignored.The FEM strategy to calculate nonlinear dynamic response of the truss stuctures takes a lot of CPU time and is inconvenient for nonlinear mechanism analsis.Considering the geometric nonlineaity of the truss,this thesis introduces von Karman nonlinear strain-displacement relationship based on the proposed equivalent beam model and establishes equivalent nonlinear beam model for the nonlinear truss structure with two pinned ends.Employing the Hamilton principle and Galerkin method,the reduced and discrete model of the equivalent beam model is obtained.The nonlinear static deformation and dynamic response of the equivalent model and truss structure under different external loading are investigated to validate the effectiveness of the proposed equivalent nonlinear beam model.Computing time of the equivalent nonlinear beam model is compared with that of the full-scale finite element model in ANSYS so as to highlight the significant less computational cost of the proposed equivalent model.Finally,the influence of mode order selecting on the dynamic response of the system is discussed on the basis of the frequency-response curves of the equivalent nonlinear beam model.The degree of freedom of the FE model for the large truss structure is too large to design control system conveniently.On the contrary,the modal approach can significantly reduce the number of degree of freedom for system and facilitates the design of low order control law.This thesis firstly establishes the equivalent beam model and obtains its analytical mode shape expressions.Then,the reduced and discrete model of the equivalent beam model is achieved.The LQR vibration controller of the equivalent beam model is designed for periodic excitation and pulsing disturbance.Simultaneously,the vibration control law is applied to the original truss system employing ANSYS finite element simulation software.Comparisons of numerical simulation demonstrate that the designed vibration controller can suppress effectively the vibration of original truss structure based on the equivalent beam model.更多还原