固体火箭发动机药柱的粘弹性不确定结构分析
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摘要
固体火箭发动机药柱是由固体推进剂整体浇铸而成的，既为发动机提供燃料，又是重要的结构组成部分，其安全性不容忽视。固体推进剂具有粘弹性和近似不可压缩性，其材料参数和外激励过程通常不能精确给出。因此，进行固体火箭发动机药柱的不确定结构分析具有重要的理论价值和工程实际意义。本文重点建立了固体药柱不确定结构分析的粘弹性随机有限元方法和非概率方法，研究了药柱的结构可靠度计算问题以及不同载荷作用下具有随机参数药柱的结构响应问题。主要研究内容如下：
从Herrmann变分原理出发，基于适用于近似不可压缩材料的粘弹性本构关系，利用增量法处理遗传积分，发展了一种粘弹性增量有限元方法，所需存储空间较少，参数矩阵形式简单，求解过程只需要形成一次刚度矩阵，适用于药柱的结构计算，奠定了确定性分析基础。
基于随机场的局部平均法以及随机过程的Karhunen-Loeve分解理论，通过一阶随机摄动方法建立了考虑材料近似不可压缩的粘弹性随机有限元公式，由相关结构分解减少计算量，分析了各结构随机响应量之间的关系，给出了数字特征的计算方法，研究了粘弹性随机结构的Monte Carlo模拟验证方法。尽管粘弹性本构关系具有时间相依性，其随机摄动格式并不存在“长期项”的影响。
建立了粘弹性随机分析的虚功原理，可以统一考虑各类随机因素，理论基础更加完善。利用向量随机场模型和二阶随机摄动方法，给出了相应的粘弹性随机有限元列式。能够考虑参数的相关性，更加接近于Prony级数的参数测试过程。所适用的参数变异范围更广。
建立了基于Total Lagrangian法的随机虚功方程，能够同时考虑粘弹性、大变形以及随机性的影响，导出了非线性粘弹性随机有限元平衡方程，分析了其具体迭代求解方法。适于固体药柱模量较低、变形较大的特点。
采用凸集合模型表示药柱参数的不确定性，在粘弹性有限元摄动分析基础上进行了结构的响应区间计算，讨论了概率方法和集合理论模型得到的响应区间之间的关系，利用响应曲面法进行了区间的二次估计。
基于粘弹性随机有限元研究了药柱结构的瞬时可靠度、首次超越可靠度以及模糊可靠度问题。从工程实际出发，分析了不确定参数对药柱结构的影响。粘弹性随机有限元方法既可以提供可靠度指标，又能够根据给定可靠度或安全系数对参数的测试精度提出要求。
国防科学技术大学研究生院学位论文
总之，本文较成功地建立了固体火箭发动机药柱的粘弹性不确定结构分析理
论，在方法和应用上都取得了一定的进展，为进一步研究和分析实际药柱结构奠
定了基础。
Solid Rocket Motor (SRM) grain is casted integrally from solid propellants. It is one of the important structural components and also provides fuels for SRM. The reliability of SRM grain should be paid great attention. However, solid propellant is viscoelastic and nearly-incompressible, and the material parameters and load process can't be given accurately in general. So it has important value both on theory and practical application to perform uncertain structural analysis for SRM grain. The main purpose of this paper is to establish viscoelastic stochastic finite element method and non-probabilistic method for uncertain structural analysis of SRM grain. Several reliability models are analyzed and structural response of SRM grain with random parameters under different loads is investigated. The main work and achievements are summarized as follows:
Based on Herrmann variational principle and the nearly-incompressible viscoelastic constitutive relation, a new type of viscoelastic incremental finite element method is presented. In this method, incremental arithmetic is applied to solving the hereditary integrals. It needs less memory space, has simple parameter matrix, and calculates stiffness matrix only once. The method is suitable for structural analysis of SRM grain, and furthermore, provides the basis for viscoelastic uncertain analysis.
A new type of viscoelastic stochastic finite element method is established using first-order perturbation theory based on local averaging method of random field and Karhunen-Loeve expansion theory of random process. The amount of computations is greatly reduced by transforming correlated random variables to a set of uncorrelated random variables. The relations of different random response variables are analyzed and Monte Carlo simulations for viscoelastic stochastic structures are investigated. The application of stochastic perturbation method to viscoelastic problems doesn't result in the emergence of undesirable secular terms although the viscoelastic constitutive relation is time-dependent.
The stochastic principle of virtual work is formulated for viscoelastic structural analysis. The innovated principle has solid theoretical basis and allows incorporation of different kinds of random factors. By using vector random field model and second-order perturbation method, viscoelastic stochastic finite element formulation is developed, which can take the correlation of random parameters into account and is suitable for stochastic structural analysis with larger variance of the uncertainties.
The stochastic equation of virtual work based on Total Lagrangian method is developed to analyze the influences of vicoelasticity, large deformation and randomness simultaneously. The nonlinear equilibrium equations of viscoelastic
stochastic FEM are derived. The full Newton-Raphson method is used to get the numerical solutions. This developed method is suitable for SRM grain with low modulus and large deformation.
Convex models are defined to describe the uncertainties of parameters in SRM grain. Perturbation method is used to predict the variability of structural response. By comparing with stochastic structural analysis, relations between the two methods have been found. The response surface method is used to obtain the second order estimates of the calculated results bound.
By means of viscoelastic stochastic finite element method, the instantaneous reliability, the first passage reliability and fuzzy reliability of SRM grain are investigated. The influences of uncertain parameters on SRM grain structures are studied. Viscoelastic stochastic FEM can not only calculate reliability index, but also determine the accuracy requirement for parametric testing according to given reliability.
In conclusion, viscoelastic uncertain structural analysis methods have been successfully established for SRM grain. Some developments have been gained both in methods and applications. The research in this paper lays a foundation for further investigation of practical SRM grain structures.
从Herrmann变分原理出发，基于适用于近似不可压缩材料的粘弹性本构关系，利用增量法处理遗传积分，发展了一种粘弹性增量有限元方法，所需存储空间较少，参数矩阵形式简单，求解过程只需要形成一次刚度矩阵，适用于药柱的结构计算，奠定了确定性分析基础。
基于随机场的局部平均法以及随机过程的Karhunen-Loeve分解理论，通过一阶随机摄动方法建立了考虑材料近似不可压缩的粘弹性随机有限元公式，由相关结构分解减少计算量，分析了各结构随机响应量之间的关系，给出了数字特征的计算方法，研究了粘弹性随机结构的Monte Carlo模拟验证方法。尽管粘弹性本构关系具有时间相依性，其随机摄动格式并不存在“长期项”的影响。
建立了粘弹性随机分析的虚功原理，可以统一考虑各类随机因素，理论基础更加完善。利用向量随机场模型和二阶随机摄动方法，给出了相应的粘弹性随机有限元列式。能够考虑参数的相关性，更加接近于Prony级数的参数测试过程。所适用的参数变异范围更广。
建立了基于Total Lagrangian法的随机虚功方程，能够同时考虑粘弹性、大变形以及随机性的影响，导出了非线性粘弹性随机有限元平衡方程，分析了其具体迭代求解方法。适于固体药柱模量较低、变形较大的特点。
采用凸集合模型表示药柱参数的不确定性，在粘弹性有限元摄动分析基础上进行了结构的响应区间计算，讨论了概率方法和集合理论模型得到的响应区间之间的关系，利用响应曲面法进行了区间的二次估计。
基于粘弹性随机有限元研究了药柱结构的瞬时可靠度、首次超越可靠度以及模糊可靠度问题。从工程实际出发，分析了不确定参数对药柱结构的影响。粘弹性随机有限元方法既可以提供可靠度指标，又能够根据给定可靠度或安全系数对参数的测试精度提出要求。
国防科学技术大学研究生院学位论文
总之，本文较成功地建立了固体火箭发动机药柱的粘弹性不确定结构分析理
论，在方法和应用上都取得了一定的进展，为进一步研究和分析实际药柱结构奠
定了基础。
Solid Rocket Motor (SRM) grain is casted integrally from solid propellants. It is one of the important structural components and also provides fuels for SRM. The reliability of SRM grain should be paid great attention. However, solid propellant is viscoelastic and nearly-incompressible, and the material parameters and load process can't be given accurately in general. So it has important value both on theory and practical application to perform uncertain structural analysis for SRM grain. The main purpose of this paper is to establish viscoelastic stochastic finite element method and non-probabilistic method for uncertain structural analysis of SRM grain. Several reliability models are analyzed and structural response of SRM grain with random parameters under different loads is investigated. The main work and achievements are summarized as follows:
Based on Herrmann variational principle and the nearly-incompressible viscoelastic constitutive relation, a new type of viscoelastic incremental finite element method is presented. In this method, incremental arithmetic is applied to solving the hereditary integrals. It needs less memory space, has simple parameter matrix, and calculates stiffness matrix only once. The method is suitable for structural analysis of SRM grain, and furthermore, provides the basis for viscoelastic uncertain analysis.
A new type of viscoelastic stochastic finite element method is established using first-order perturbation theory based on local averaging method of random field and Karhunen-Loeve expansion theory of random process. The amount of computations is greatly reduced by transforming correlated random variables to a set of uncorrelated random variables. The relations of different random response variables are analyzed and Monte Carlo simulations for viscoelastic stochastic structures are investigated. The application of stochastic perturbation method to viscoelastic problems doesn't result in the emergence of undesirable secular terms although the viscoelastic constitutive relation is time-dependent.
The stochastic principle of virtual work is formulated for viscoelastic structural analysis. The innovated principle has solid theoretical basis and allows incorporation of different kinds of random factors. By using vector random field model and second-order perturbation method, viscoelastic stochastic finite element formulation is developed, which can take the correlation of random parameters into account and is suitable for stochastic structural analysis with larger variance of the uncertainties.
The stochastic equation of virtual work based on Total Lagrangian method is developed to analyze the influences of vicoelasticity, large deformation and randomness simultaneously. The nonlinear equilibrium equations of viscoelastic
stochastic FEM are derived. The full Newton-Raphson method is used to get the numerical solutions. This developed method is suitable for SRM grain with low modulus and large deformation.
Convex models are defined to describe the uncertainties of parameters in SRM grain. Perturbation method is used to predict the variability of structural response. By comparing with stochastic structural analysis, relations between the two methods have been found. The response surface method is used to obtain the second order estimates of the calculated results bound.
By means of viscoelastic stochastic finite element method, the instantaneous reliability, the first passage reliability and fuzzy reliability of SRM grain are investigated. The influences of uncertain parameters on SRM grain structures are studied. Viscoelastic stochastic FEM can not only calculate reliability index, but also determine the accuracy requirement for parametric testing according to given reliability.
In conclusion, viscoelastic uncertain structural analysis methods have been successfully established for SRM grain. Some developments have been gained both in methods and applications. The research in this paper lays a foundation for further investigation of practical SRM grain structures.
引文
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