基于EEP法的平面变截面杆件自由振动自适应分析
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摘要
变截面杆件在各工程领域中有着广泛应用,对其自由振动做高效精确的分析具有重要意义。随着数值计算的发展,自适应求解功能正成为各种数值方法所竞相追求的目标。本文基于具有最佳超收敛阶的超收敛算法——单元能量投影(Element Energy Projection,简称EEP)法凝聚格式,对平面变截面杆件的自由振动问题建立了一套自适应分析方法,并成功将其推广应用至求解一般的2阶Sturm-Liouville(简称SL)问题和4阶SL问题中。
本文首先以变截面杆件的轴向自由振动问题为模型,通过一维Ritz有限元C~0问题EEP法凝聚格式的自适应算法求解杆件单元的动力形函数,探索和验证了基于EEP法的自适应算法在振动问题求解中的可行性。在这一基础上,通过对动力刚度法与精确有限元法等价性的深入认识,本文开展了下列工作:
第一,将EEP法凝聚格式的自适应算法和Wittrick-Williams算法相结合,建立了杆件动力分析通用的自适应求解方案、实施策略和具体算法,并用于变截面杆件轴向自由振动和Euler梁横向自由振动的求解中。
第二,推导了变截面Timoshenko曲梁自由振动的微分控制方程。基于2阶常微分方程组EEP法凝聚格式,将上述自适应求解思想用于求解变截面Timoshenko直梁横向自由振动和曲梁自由振动的常微分方程组特征值问题。
第三,通过对2阶和4阶SL问题力学模型的分析,提出了SL问题中边界条件、负特征值问题的处理方案,并将杆件自由振动的自适应求解方案推广至SL问题的求解中。
本文编制了上述问题的自适应求解程序,并对各种问题中代表性的算例进行了分析。文中的理论研究和数值试验表明,本文方法不依赖于用户提供的初始网格,程序能自动给出满足用户误差要求的频率(特征值)和振型(特征函数),且任一点的位移解和内力解均具有和单元结点位移解相当的精度。本文方法几乎克服了其它现有方法的缺陷,具有精确、稳定、高效、实施方便等特点。本文自适应算法的思想为求解变截面杆系结构自由振动问题及一般的常微分方程组特征值问题等开辟了新的途径。
It is of great significance to make efficient and accurate analysis to the free vibration of non-uniform members, which have wide applications in many fields of engineering. With the development of numerical computation, adaptive analysis becomes the goal of various numerical methods. Based on the condensed scheme of Element Energy Projection (EEP) method with optimal super-convergent order, the dissertation proposed an adaptive solution for the free vibration of non-uniform members. The above method is further applied to general 2nd and 4th order Sturm- Liouville eigenvalue problems successfully.
Taking the axial free vibration of non-uniform members as the model problem, the dissertation first presented the adaptive analysis, which is used in one-dimensional C 0 problem based on EEP condensed scheme, to get dynamic shape functions of elements. The above analysis is used to check the feasibility of the method in free vibration problems. Based on the in-depth understanding of equivalence between dynamic stiffness method (DSM) and the exact finite element method (FEM), the dissertation carried out the following work:
Firstly, the self-adaptive method based on EEP condensed scheme and the Wittrick-Williams algorithm were creatively combined. The dissertation proposed a generalized self-adaptive method, implementation strategy and computational algorithm in dynamic analysis of non-uniform members. Then the method was applied to solve axial free vibration of non-uniform members and flexural free vibration of non-uniform Bernoulli-Euler beams.
Secondly, the governing differential equations of free vibration of non-uniform Timoshenko curved beams was derived. The self-adaptive method to solve the flexural free vibration of non-uniform Timoshenko beams and the free vibration of non-uniform Timoshenko curved beams, which is a kind of eigenvalue problems for ordinary differential equations, was also constructed respectively.
Finally, through the analysis of mechanics models of the 2nd and 4th SL problems, the dissertation proposed a way to deal with boundary conditions and negative eigenvalues and extended the self-adaptive method of free vibration to solve the 2nd and 4th SL problems.
Computer programs have been developed for self-adaptive analysis, and representative numerical examples for various problems were analyzed. The theoretical research and numerical tests show that the method is independent of the initial mesh provided by users, not only can the program produce the frequency (eigenvalues) and vibration mode (eigenfucntion) which are satisfied the error tolerance automatically, but also the displacements and the internal forces of any point gain the equivalent numerical precision with the displacements at the element nodes. With this method, most shortcomings in other existing methods are overcome. A large number of numerical tests prove this algorithm to be accurate, stable, efficient and convenient. The concept of the adaptive method in this dissertation provides a new way to solve the free vibration of skeletal structures constructed by non-uniform members and eigenvalue problems for ordinary differential equations.
本文首先以变截面杆件的轴向自由振动问题为模型,通过一维Ritz有限元C~0问题EEP法凝聚格式的自适应算法求解杆件单元的动力形函数,探索和验证了基于EEP法的自适应算法在振动问题求解中的可行性。在这一基础上,通过对动力刚度法与精确有限元法等价性的深入认识,本文开展了下列工作:
第一,将EEP法凝聚格式的自适应算法和Wittrick-Williams算法相结合,建立了杆件动力分析通用的自适应求解方案、实施策略和具体算法,并用于变截面杆件轴向自由振动和Euler梁横向自由振动的求解中。
第二,推导了变截面Timoshenko曲梁自由振动的微分控制方程。基于2阶常微分方程组EEP法凝聚格式,将上述自适应求解思想用于求解变截面Timoshenko直梁横向自由振动和曲梁自由振动的常微分方程组特征值问题。
第三,通过对2阶和4阶SL问题力学模型的分析,提出了SL问题中边界条件、负特征值问题的处理方案,并将杆件自由振动的自适应求解方案推广至SL问题的求解中。
本文编制了上述问题的自适应求解程序,并对各种问题中代表性的算例进行了分析。文中的理论研究和数值试验表明,本文方法不依赖于用户提供的初始网格,程序能自动给出满足用户误差要求的频率(特征值)和振型(特征函数),且任一点的位移解和内力解均具有和单元结点位移解相当的精度。本文方法几乎克服了其它现有方法的缺陷,具有精确、稳定、高效、实施方便等特点。本文自适应算法的思想为求解变截面杆系结构自由振动问题及一般的常微分方程组特征值问题等开辟了新的途径。
It is of great significance to make efficient and accurate analysis to the free vibration of non-uniform members, which have wide applications in many fields of engineering. With the development of numerical computation, adaptive analysis becomes the goal of various numerical methods. Based on the condensed scheme of Element Energy Projection (EEP) method with optimal super-convergent order, the dissertation proposed an adaptive solution for the free vibration of non-uniform members. The above method is further applied to general 2nd and 4th order Sturm- Liouville eigenvalue problems successfully.
Taking the axial free vibration of non-uniform members as the model problem, the dissertation first presented the adaptive analysis, which is used in one-dimensional C 0 problem based on EEP condensed scheme, to get dynamic shape functions of elements. The above analysis is used to check the feasibility of the method in free vibration problems. Based on the in-depth understanding of equivalence between dynamic stiffness method (DSM) and the exact finite element method (FEM), the dissertation carried out the following work:
Firstly, the self-adaptive method based on EEP condensed scheme and the Wittrick-Williams algorithm were creatively combined. The dissertation proposed a generalized self-adaptive method, implementation strategy and computational algorithm in dynamic analysis of non-uniform members. Then the method was applied to solve axial free vibration of non-uniform members and flexural free vibration of non-uniform Bernoulli-Euler beams.
Secondly, the governing differential equations of free vibration of non-uniform Timoshenko curved beams was derived. The self-adaptive method to solve the flexural free vibration of non-uniform Timoshenko beams and the free vibration of non-uniform Timoshenko curved beams, which is a kind of eigenvalue problems for ordinary differential equations, was also constructed respectively.
Finally, through the analysis of mechanics models of the 2nd and 4th SL problems, the dissertation proposed a way to deal with boundary conditions and negative eigenvalues and extended the self-adaptive method of free vibration to solve the 2nd and 4th SL problems.
Computer programs have been developed for self-adaptive analysis, and representative numerical examples for various problems were analyzed. The theoretical research and numerical tests show that the method is independent of the initial mesh provided by users, not only can the program produce the frequency (eigenvalues) and vibration mode (eigenfucntion) which are satisfied the error tolerance automatically, but also the displacements and the internal forces of any point gain the equivalent numerical precision with the displacements at the element nodes. With this method, most shortcomings in other existing methods are overcome. A large number of numerical tests prove this algorithm to be accurate, stable, efficient and convenient. The concept of the adaptive method in this dissertation provides a new way to solve the free vibration of skeletal structures constructed by non-uniform members and eigenvalue problems for ordinary differential equations.
引文
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