不确定性结构的动力学分析

发布时间：2018-06-06 23:16

本文选题：泛灰数学 + 柔性梁　；参考：《西安电子科技大学》2015年博士论文

【摘要】：随机模型、模糊模型和区间模型是研究不确定性结构问题常用的三种数学模型。本文针对运用区间分析进行动力学分析时存在的计算结果偏于保守的缺点,引入泛灰数学方法进行区间分析,并提出了一种改进的泛灰数除法运算规则,通过算例验证了改进运算规则的正确性;同时,为了使现有柔性梁系统的动力学模型更符合实际工况,本文把参数的随机性引入到柔性梁系统中,分析了随机参数平面柔性梁系统、考虑附加质量的平面柔性梁系统、空间柔性梁系统的动力响应问题和动力特性问题,计算结果表明随机参数的分散性对柔性梁的动力响应及动力特性的影响不可忽视。主要内容如下:第一部分研究了基于泛灰数学的广义瑞利商算法。介绍了泛灰数和区间数的相关概念和运算法则,在区间数与泛灰数转化规则的基础上,通过数学算例说明泛灰运算具有区间分析功能,且相对于区间分析具有一定的优越性;将结构系统中的不确定参数用泛灰数表示,根据广义瑞利商的性质和对区间特征值的单调性分析,提出一种基于泛灰数学的广义瑞利商算法。以区间弹簧质量系统和多层框架结构为例说明了该方法计算简单、准确可靠,有一定的工程应用价值。第二部分研究了不确定链式结构动力特性的泛灰分析方法。通过对泛灰数四则运算规则的分析,指出泛灰数除法运算存在的缺陷,提出一种改进的泛灰数除法运算规则。基于传递矩阵法和泛灰数学方法,推导了链式结构参数具有区间不确定性时系统关于固有频率的非线性泛灰方程,提出一种区间进退算法对该方程进行求解。通过算例验证了模型的合理性,以及求解方法的有效性。第三部分分别研究了含随机参数的平面柔性梁系统、考虑附加质量的平面柔性梁系统的随机动力响应特性。将不确定性问题引入到平面柔性梁问题中,考虑系统物理参数和几何参数的随机性,采用假设模态法和拉格朗日方程建立了计及动力刚化项的旋转柔性梁随机动力学模型,利用基于高效回归法的多项式混沌法和伽辽金法将完全隐式的随机微分方程组转化为完全隐式的纯微分方程组,求解方程组得到柔性梁变形位移响应的数字特征。最后,以随机参数柔性梁系统为例,获得其动力响应统计意义下的解,通过与蒙特卡洛模拟结果比较,验证文中方法的正确性和有效性。算例结果表明部分随机参数的分散性对柔性梁的动力响应的影响不可忽视,利用含随机参数的动力学模型能客观地反映出柔性梁的动力学行为。第四部分研究了空间柔性梁系统的随机动力响应特性。基于假设模态法和虚功原理建立了计及动力刚化项的空间柔性梁随机动力学模型,利用单项式容积法作为多项式混沌的配点和伽辽金法将完全隐式的随机微分方程组转化为完全隐式的纯微分方程组,通过回归法得到空间柔性梁变形位移响应的数字特征。最后,以随机参数空间柔性梁系统为例,获得其动力响应统计意义下的解,通过与蒙特卡洛结果比较,验证文中方法的正确性和高效性。第五部分对随机参数柔性梁系统的动力特性进行了研究。分别对参数具有随机性的平面柔性梁系统、考虑附加质量的平面柔性梁系统、空间柔性梁系统的横向弯曲振动固有频率的随机特性进行研究。对柔性梁一次近似耦合随机动力学方程进行无量纲化,利用随机因子法和蒙特卡洛法得到系统的无量纲随机固有频率的随机特性。研究了各无量纲参数对系统固有频率随机特性的影响。
[Abstract]:The stochastic model, the fuzzy model and the interval model are three kinds of mathematical models to study the uncertain structure problem. In this paper, in this paper, the calculation results of the dynamic analysis in the interval analysis are partial to the conservative shortcoming, the generalized grey mathematical method is introduced to the interval analysis, and an improved operation rule of the pan grey number division is proposed. At the same time, in order to make the dynamic model of the existing flexible beam system more practical, this paper introduces the randomness of the parameters to the flexible beam system, analyzes the plane flexible beam system with random parameters, considering the plane flexible beam system with additional mass and the dynamics of the space flexible beam system. The results of the response and dynamic characteristics show that the influence of the dispersion of the random parameters on the dynamic response and dynamic characteristics of the flexible beam can not be ignored. The main contents are as follows: in the first part, the generalized Rayleigh quotient algorithm based on Pan grey mathematics is studied. The related concepts and operations of the generalized grey number and interval numbers are introduced, and the interval numbers are introduced. On the basis of the transformation rule of the pan grey number, a mathematical example shows that the pan grey operation has the function of interval analysis and is superior to the interval analysis. The uncertainty parameters in the structural system are expressed by the pan grey number. Based on the properties of the generalized Rayleigh quotient and the monotonicity analysis of the interval eigenvalues, a kind of Pan grey mathematics is proposed. The generalized Rayleigh quotient algorithm, taking the interval spring mass system and the multi-layer frame structure as an example, shows that the method is simple, accurate and reliable, and has certain engineering application value. The second part studies the pan grey analysis method of the dynamic characteristics of the uncertain chain structure. A modified universal grey division algorithm is proposed. Based on the transfer matrix method and the pan grey mathematical method, the nonlinear generalized grey equation for the natural frequency of the chain structure parameters with interval uncertainty is derived, and an interval algorithm is proposed to solve the equation. A numerical example is given to verify the model. The third part studies the plane flexible beam system with random parameters, considering the random dynamic response characteristics of the plane flexible beam system with additional mass, and introduces the uncertainty problem to the plane flexible beam problem, considering the randomness of the physical parameters and geometric parameters of the system, and using the false method. The model method and Lagrange equation are set up to establish the stochastic dynamic model of the rotating flexible beam with the dynamic stiffening term. By using the polynomial chaos method and Galerkin method based on high efficiency regression method, the completely implicit differential equations are converted into completely implicit differential equations, and the deformation displacement response of the flexible beam is obtained by solving the equations. Finally, a random parameter flexible beam system is taken as an example to obtain the solution in the statistical significance of the dynamic response. By comparing with the Monte Carlo simulation results, the correctness and effectiveness of the method are verified. The results of the calculation show that the dispersion of some random parameters can not be ignored and the random parameters are not ignored. The dynamics model of the number can objectively reflect the dynamic behavior of the flexible beam. The fourth part studies the random dynamic response characteristic of the space flexible beam system. Based on the hypothesis modal method and the virtual work principle, the stochastic dynamic model of the space flexible beam with the dynamic stiffening term is established, and the monomial volume method is used as the matching of the polynomial chaos. Point and Galerkin method convert completely implicit stochastic differential equations into completely implicit pure differential equations. The numerical characteristics of deformation and displacement responses of space flexible beams are obtained by regression method. Finally, a random parameter space flexible beam system is taken as an example to obtain the solution under the statistical significance of its dynamic response, by comparing with the Monte Carlo results, The correctness and efficiency of the method are verified. In the fifth part, the dynamic characteristics of the flexible beam system with random parameters are studied. The random characteristics of the plane flexible beam system with random parameters, the plane flexible beam system with additional mass and the natural frequency of the lateral flexural vibration of the space flexible beam system are studied. The first approximation coupled stochastic dynamic equations of the flexible beam are dimensionless, and random factor method and Monte Carlo method are used to obtain the random characteristic of the dimensionless random natural frequency of the system. The influence of the dimensionless parameters on the stochastic characteristic of the natural frequency of the system is studied.
【学位授予单位】：西安电子科技大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：TB122

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