考虑参数不确定性的结构系统振动研究

发布时间：2019-01-04 11:20

【摘要】：本文主要针对具有不确定性因素结构系统的振动统计特性进行研究分析,基于正交多项式函数系的性质以及多变量函数降维分解等方法,用于解决实际工程和科学领域中遇到的随机性问题,并将所得随机响应的概率统计特征与直接Monte Carlo模拟方法所得结果进行对比验证。首先,介绍了埃尔米特、勒让德等正交多项式的基本概念及性质,它们是对响应函数逼近的有利工具。基于任意连续可微的多元函数维分解算法,针对结构系统的随机响应,在系统各不确定性因素均服从各自相应且相互独立的Gauss分布条件下,利用Fourier-Hermite多项式展开,以及通过广义模型降维、多重Gauss-Hermite数值积分的方法确定各个展开系数,以获得要求随机响应的显式正交多项式函数逼近形式,并通过嵌入局部Monte Carlo模拟方法分析其概率统计特征,最后通过误差公式求得其与直接Monte Carlo模拟方法的前四阶原点矩误差,以更直观的方式进行对比。其次,在实际结构动力系统中,固有频率作为结构动力特征参数,是系统设计、结构分析和稳定性、敏感度分析中一个关键参数,而考虑系统不确定性时其具有随机性特征。针对某一具有三自由度且忽略阻尼存在的弹簧质量系统,对其各阶固有频率进行数值仿真模拟,结果表明:“黑匣子”结构方法的各阶固有频率统计结果均优于隐式函数表达结构方法;且随着考虑系统不确定性参数变量元数的增加,其各阶固有频率统计特征将更加符合直接Monte Carlo模拟求解结果,但计算成本也将随之呈多项式增长,故应当根据误差分析选择适当随机变量元数,以在满足求解精度的同时尽可能地降低计算工作量。最后,对实际工程结构中广泛应用的板壳结构单元进行参数化建模,在两侧自由、左边固支-右边弹簧支撑的边界条件下,受到单点简谐力激励起原点振动响应。基于上述分析方法,针对其在激励点处z方向的稳态振动位移响应进行分析,获得其系统随机位移响应的概率统计特征;并通过网格单元细化标准对板结构网格进行逐步细化,分析变弹性支撑随机边界条件对随机位移响应的影响。仿真结果表明:提出的方法可获得与直接Monte Carlo模拟较一致的分析结果,能够获得随机板结构振动响应统计特征,并基于离散刚度边界的有限单元法网格细化,预测连续刚度边界的随机板结构振动响应统计特征趋于某一确定分布。
[Abstract]:In this paper, the statistical characteristics of vibration of structural systems with uncertain factors are studied and analyzed, based on the properties of orthogonal polynomial function system and the method of dimensionality reduction of multivariable functions. It is used to solve the randomness problems encountered in practical engineering and science, and the probabilistic and statistical characteristics of the random responses are compared with the results obtained by direct Monte Carlo simulation. Firstly, the basic concepts and properties of Hermite and Legendre orthogonal polynomials are introduced, which are favorable tools for approximation of response functions. Based on the dimensionality decomposition algorithm of arbitrary continuous differentiable multivariate functions, the Fourier-Hermite polynomial is used to expand the random response of the structural system under the condition that all the uncertain factors of the system are based on the corresponding and independent Gauss distribution. By reducing the dimension of the generalized model and using the method of multiple Gauss-Hermite numerical integration, each expansion coefficient is determined to obtain the approximation form of explicit orthogonal polynomial function which requires random response. The probabilistic and statistical characteristics of the method are analyzed by embedding local Monte Carlo simulation method. Finally, the error of the first four origin moments of the direct Monte Carlo simulation method is obtained by error formula, which is compared in a more intuitive way. Secondly, natural frequency is a key parameter in system design, structural analysis and stability, sensitivity analysis, as a structural dynamic characteristic parameter in actual structural dynamic system, and it has stochastic characteristics when considering the uncertainty of the system. For a spring mass system with three degrees of freedom and neglecting damping, the numerical simulation of its natural frequencies is carried out. The results show that the statistical results of the natural frequencies of the "black box" structure method are better than the implicit function expression method. With the increase of variable number of uncertain parameters, the statistical characteristics of natural frequencies of each order will be more consistent with the results of direct Monte Carlo simulation, but the computational cost will also increase with polynomial. Therefore, it is necessary to select the appropriate number of random variables according to the error analysis in order to reduce the calculation workload while satisfying the accuracy of the solution. Finally, parameterized modeling of plate-shell structural elements which are widely used in practical engineering structures is carried out. Under the boundary conditions of free on both sides and supported by spring on the left and right, the vibration response of the origin is caused by a simple harmonic force at a single point. Based on the above analysis method, the steady-state vibration displacement response in z direction at the excitation point is analyzed, and the probability and statistical characteristics of the random displacement response of the system are obtained. The mesh of plate structure is refined step by mesh element refinement criterion, and the influence of random boundary conditions of variable elastic braces on the response of random displacement is analyzed. The simulation results show that the proposed method can obtain the analytical results consistent with the direct Monte Carlo simulation and obtain the statistical characteristics of the vibration response of the stochastic plate structures. The finite element method based on the discrete stiffness boundary is used to refine the meshes. The statistical characteristics of the vibration response of stochastic plate structures with continuous stiffness boundary tend to a certain distribution.
【学位授予单位】：东北电力大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O327

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