摘要
基于固体物理周期结构的基本理论,将其应用于结构隔震来研究地震动波在周期结构中的传播。首先构造复合周期钢结构模型,以两两单元之间相位差为参数:设钢结构中至少含2个有方向的周期单元,将这些单元两两黏合。首先固定第一个格子,按照黏合映射与方向复合的等距映射,它和第二个格子形成两格子周期结构,且其中2个周期结构方向之间有正的相位差。这2个格子向两端延伸,构成一维复合周期结构,且包含不少于2个有方向的周期结构。在一维复合周期结构中,每个格子再向平面内、空间内按照黏合映射与方向复合的等距映射延伸,组成二维、大于二维的复合周期结构。黏合映射与方向复合构成两两黏合后周期结构之间公共部分的映射,将地震动波从第一个单元传播到第二个单元,再依次传播到整个复合周期钢结构。接着计算地震动波在复合周期结构中的响应,影响参数有:波数、黏合映射数、周期结构的尺度和相位差。假设响应是线性的,首先分别讨论单个参数影响,最后求和得到复合周期结构总响应。由于地震动波具有随机性,服从给定的随机微分方程,结构响应也是可能取值的集合。故先讨论2个周期单元情形、一维情形的地震动响应。在两格子复合周期钢结构中,先在第一个格子输入给定的随机微分方程、初始条件和边界条件,应用集值随机过程和鞅理论计算,得到第一个格子的响应;再由相位差和三角函数性质,计算第二个格子的响应、一维结构中每一个格子的响应。最后实施结构地震动响应的Markov控制,选择最优相位角,实现一类多项式函数响应目标:根据周期单元响应的集值随机过程性质,表示为扩散过程,由它的无穷小生成元构造对应的Dirichlet-Poisson方程,进而构造Hamilton-Jacobi-Bellman(HJB)方程,解得最优相位角,代回Dirichlet-Poisson方程,再求解得到控制结果。该响应随地震动波传播时间、初始幅值增大而降低。应用Monte Carlo模拟给定的随机地震动波,生成平稳分布,再采样。该采样序列分布服从该平稳分布,计算采样序列和Markov控制的方差,再比较随机地震动波示性函数和Markov控制的期望,表明Markov控制序列的波动性小于采样序列,Markov控制也降低了地震动响应的期望值。该方法实现了复合周期钢结构地震动波响应的随机过程表示及Markov控制,为复合周期钢结构地震动波响应随机预测提供参考。
Fundamentals based on periodic structure in Solid-State Physics are applied to seismic isolation and propagation in periodic structure.Firstly,a compound periodic steel structure parameterized by its phase differences between cells is constructed:It is supposed that a steel structure contains no less than 2 orientable periodic cells,and these cells are glued together one by one.Beginning from the fixed first cell,by isometric mapping composed of gluing and orientation maps,it aligned with the 2nd cell forms a 2-cell periodic structure with positive phase difference between their orientations,then the 2-cell structure extends at its both ends into an 1-dimensional periodic structure containing at least 2 orientable periodic substructures.In this 1-dimensional compound periodic structure,each cell further extends on plane,or into space,according to isometric mapping composed of gluing and orientation maps,and shapes a 2-dimensioanal,and those compound periodic structures of dimensions bigger than 2.Composition of gluing maps with orientations plays the mapping role at those common parts of periodic structures glued together one by one,propagates seismic wave from the beginning cell to the next cell,in turn seismic wave ranges through the integrated compound periodic structure.Secondly,seismic response parameterized by wave number,number of gluing mappings,scale of the structure,and phase differences in the structures is computed in compound periodic structure.By linear response assumption,response with each of parameters is computed respectively at first,then total response of compound periodic structure is obtain from their summation.Seismic wave is stochastic,and obeys a given stochastic differential equation,and then structural response is also set of values with probability.Seismic responses in 2-cell and 1-dimensional periodic structure are discussed at first.In 2-cell compound periodic steel structure,the given stochastic differential equation is input at its first cell,under certain starting condition and boundary condition,and with the help of set-valued stochastic process and martingale knowledge,seismic response in the first cell is calculated;further by phase difference and trigonometric function operations,seismic responses in the second cell of 2-cell structure and each cell in 1-dimensional structure are calculated.Finally,Markov controlling to structural seismic response is performed to select the optimal phase angle,and realize a sort of polynomial response goal:according to the set-valued stochastic process property of the response in periodic cell,it is represented by a diffusion process,and its infinitesimal generator is used to construct the corresponding Dirichlet-Poisson equation,further the corresponding Hamilton-Jacobi-Bellman(HJB)equation is constructed to solve the optimal phase angle,substituted back into the Dirichlet-Poisson equation to solve the controlling result.The result decreases as propagating time and initial amplitude of seismic wave increase.Simulating the given stochastic seismic wave by Monte Carlo method,and generating a stationary distribution,then sampling the given stochastic seismic wave by this distribution,calculation of variances of the sampling series and the Markov controlling result and comparison of expectations of indicator function of the given stochastic seismic wave and the Markov controlling result show that variance of the Markov controlling result is smaller than that of sampling series,what is more,the Markov controlling deceases expectation of the given stochastic seismic wave.Realization of both representation to seismic response in compound periodic steel structure by stochastic equation and its Markov controlling is in servitude to reference for its stochastic prediction.
作者
吴文忠
Wenzhong Wu(College of Safety Science and Engineering,Xi’an University of Science and Technology,Xi’an 710054,China)
出处
《钢结构（中英文）》
2023年第12期48-53,共6页
Steel Construction（Chinese & English）
关键词
钢结构
地震动波
复合周期结构
随机控制
steel structure
earthquake wave
compound periodic steel structure
stochastic controlling
