矩形板问题的Hamilton求解方法
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摘要
弹性矩形板是一种重要的结构元件,广泛应用于土木工程、海洋工程、航空航天以及机械工程等多个领域,其相关力学问题(弯曲、振动等)的求解一直是工程领域研究的一个重要内容,然而由于数学上的困难,该类问题的解析求解一直是一个难题。本文的工作是将该类问题导入Hamilton体系并利用辛几何方法求解典型边界条件下弹性矩形板的问题,其中包括Kirchhoff板(基于经典薄板理论)和Reissner板(基于中厚板理论)的弯曲和振动问题。
对于薄板的弯曲问题,本文从Kirchhoff薄板弯曲问题的控制方程出发,以基本力学量为对偶变量,构造出了该问题的Hamilton体系。以此为基础,再利用辛几何方法理性求解对偶方程。对于对边简支薄板,直接求出了Levy型解析解。对于对边固支的情况,以变分边界条件导出方程组来决定级数中出现的待定系数,从而得到了解析解。同时,本文还将Hamilton体系求解方法推广到各向异性的情况,建立了正交各向异性薄板弯曲问题的Hamilton体系,求解出对边简支以及对边固支正交各向异性矩形薄板的辛解析解。针对其他非对边简支矩形薄板的问题,本文提出一种基于辛几何法与叠加法结合的求解方法,作者称之为“辛—叠加方法”——该方法对于常见边界条件下的矩形板问题都是适用的。
对于中厚板的弯曲问题,从Reissner板弯曲问题的控制方程出发,首先构造出一种形式简洁的Hamilton体系,然后利用辛几何方法理性求得了对边简支Reissner板的解析解,并利用得到的解析解分析和阐释了板弯曲中的边界效应问题。与以基本力学量为对偶变量的方程相比,本文构造出的对偶方程具有形式简洁、求解方便的特点。
本文还分别将Kirchhoff板、Reissner板的自由振动问题导入Hamilton体系,理性求得了对边简支板自由振动问题的解析解。
本文的求解方法是直接从弹性矩形板的控制方程出发,将问题导入到Hamilton体系当中,然后基于辛几何方法,利用分离变量、辛本征展开等手段,得到矩形薄板、中厚板的弯曲和振动问题的解析解。由于在求解过程中不需要预先人为选取试函数(如挠度函数),而是直接以板的基本方程为起点,通过逐步的理性推导得到问题的解析解,从而使本文求解方法具有明显优于传统解析解法的优点,跳出了半逆法的限制,可以得到更多传统方法难以得到的解析解。
As the important structural elements, elastic rectangular plates are widely used in various fields such as civil engineering, mechanical engineering, ocean engineering, aeronautics and astronautics. Solution of the plate problems (bending, vibration, etc.) have been one of the important research topics in engineering. However, it is hard to obtain the analytical solutions to most of these problems till now due to the mathematical challenge. In this dissertation, the bending and vibration problems of rectangular plates, including those based on the Kirchhoff theory (thin plate theory) and Reissner theory (moderately thick plate theory), are respectively introduced into the Hamiltonian system. Accordingly, the symplectic geometry is applied to solve the problems of rectangular plates with typical boundary conditions, some of which are known as the difficulties in elasticity.
For the bending of thin plates, the Hamiltonian system is constructed from the governing equations, with the basic mechanical quantities as the symplectic variables. Then the symplectic approach is used to solve the canonical equation rationally. For the rectangular thin plates with two opposite edges simply supported, the Levy-type analytical solutions are derived; for those with two opposite edges clamped, the symplectic analytical solutions are obtained by determine the coefficients in the resultant series via the variational equations. In addition, the Hamiltonian system based solution method is extended to the bending of orthotropic plates and the sympletic analytical solutions of plates with two opposite edges simply supported or clamped are obtained. For the rectangular thin plates with complex boundary conditions such as cantilever plates, fully free plates and plates with two adjacent edges free, a symplectic superposition method is proposed, which is applicable to rectangular plates with any combinations of commonly used boundary conditions.
For the bending of moderately thick plates, a concise form of the Hamiltonian system is constructed from the governing equations. Then the symplectic approach is used to solve the bending of moderately thick rectangular plates with two opposite edges simply supported as well as to explain the boundary effects in plate bending. Compared with the canonical equation based on basic mechanical quantities, the equation derived in this dissertation is simple in form and thus easy to solve.
Free vibration problems of the thin and moderately thick plates are introduced into the Hamiltonian system and the plates with two opposite edges simply supported are rationally solved.
The solution approach presented in this dissertation starts from the governing equations of plates. In the Hamiltonian system, using the symplectic geometry, the method of separation of variables as well as symplectic eigen expansion is adopted to obtain the analytical solutions of bending and vibration of thin and moderately thick rectangular plates. It is noted that the analytical solution procedure in this dissertation is completely rational without any predetermined trial functions such as the deflections, therefore, the solution methodology prevails over the conventional ones, as represented by the semi-inverse method. As a result, we conclude that the Hamiltonian system based solution approach enables one to obtain more analytical solutions which have not been obtained by other existing methods.
对于薄板的弯曲问题,本文从Kirchhoff薄板弯曲问题的控制方程出发,以基本力学量为对偶变量,构造出了该问题的Hamilton体系。以此为基础,再利用辛几何方法理性求解对偶方程。对于对边简支薄板,直接求出了Levy型解析解。对于对边固支的情况,以变分边界条件导出方程组来决定级数中出现的待定系数,从而得到了解析解。同时,本文还将Hamilton体系求解方法推广到各向异性的情况,建立了正交各向异性薄板弯曲问题的Hamilton体系,求解出对边简支以及对边固支正交各向异性矩形薄板的辛解析解。针对其他非对边简支矩形薄板的问题,本文提出一种基于辛几何法与叠加法结合的求解方法,作者称之为“辛—叠加方法”——该方法对于常见边界条件下的矩形板问题都是适用的。
对于中厚板的弯曲问题,从Reissner板弯曲问题的控制方程出发,首先构造出一种形式简洁的Hamilton体系,然后利用辛几何方法理性求得了对边简支Reissner板的解析解,并利用得到的解析解分析和阐释了板弯曲中的边界效应问题。与以基本力学量为对偶变量的方程相比,本文构造出的对偶方程具有形式简洁、求解方便的特点。
本文还分别将Kirchhoff板、Reissner板的自由振动问题导入Hamilton体系,理性求得了对边简支板自由振动问题的解析解。
本文的求解方法是直接从弹性矩形板的控制方程出发,将问题导入到Hamilton体系当中,然后基于辛几何方法,利用分离变量、辛本征展开等手段,得到矩形薄板、中厚板的弯曲和振动问题的解析解。由于在求解过程中不需要预先人为选取试函数(如挠度函数),而是直接以板的基本方程为起点,通过逐步的理性推导得到问题的解析解,从而使本文求解方法具有明显优于传统解析解法的优点,跳出了半逆法的限制,可以得到更多传统方法难以得到的解析解。
As the important structural elements, elastic rectangular plates are widely used in various fields such as civil engineering, mechanical engineering, ocean engineering, aeronautics and astronautics. Solution of the plate problems (bending, vibration, etc.) have been one of the important research topics in engineering. However, it is hard to obtain the analytical solutions to most of these problems till now due to the mathematical challenge. In this dissertation, the bending and vibration problems of rectangular plates, including those based on the Kirchhoff theory (thin plate theory) and Reissner theory (moderately thick plate theory), are respectively introduced into the Hamiltonian system. Accordingly, the symplectic geometry is applied to solve the problems of rectangular plates with typical boundary conditions, some of which are known as the difficulties in elasticity.
For the bending of thin plates, the Hamiltonian system is constructed from the governing equations, with the basic mechanical quantities as the symplectic variables. Then the symplectic approach is used to solve the canonical equation rationally. For the rectangular thin plates with two opposite edges simply supported, the Levy-type analytical solutions are derived; for those with two opposite edges clamped, the symplectic analytical solutions are obtained by determine the coefficients in the resultant series via the variational equations. In addition, the Hamiltonian system based solution method is extended to the bending of orthotropic plates and the sympletic analytical solutions of plates with two opposite edges simply supported or clamped are obtained. For the rectangular thin plates with complex boundary conditions such as cantilever plates, fully free plates and plates with two adjacent edges free, a symplectic superposition method is proposed, which is applicable to rectangular plates with any combinations of commonly used boundary conditions.
For the bending of moderately thick plates, a concise form of the Hamiltonian system is constructed from the governing equations. Then the symplectic approach is used to solve the bending of moderately thick rectangular plates with two opposite edges simply supported as well as to explain the boundary effects in plate bending. Compared with the canonical equation based on basic mechanical quantities, the equation derived in this dissertation is simple in form and thus easy to solve.
Free vibration problems of the thin and moderately thick plates are introduced into the Hamiltonian system and the plates with two opposite edges simply supported are rationally solved.
The solution approach presented in this dissertation starts from the governing equations of plates. In the Hamiltonian system, using the symplectic geometry, the method of separation of variables as well as symplectic eigen expansion is adopted to obtain the analytical solutions of bending and vibration of thin and moderately thick rectangular plates. It is noted that the analytical solution procedure in this dissertation is completely rational without any predetermined trial functions such as the deflections, therefore, the solution methodology prevails over the conventional ones, as represented by the semi-inverse method. As a result, we conclude that the Hamiltonian system based solution approach enables one to obtain more analytical solutions which have not been obtained by other existing methods.
引文
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