小波数值方法及其在薄板结构非线性分析中的应用

发布时间：2018-06-12 02:51

本文选题：圆薄板大挠度 + 非线性振动　；参考：《兰州大学》2017年硕士论文

【摘要】：圆薄板被应用于各类工程结构之中,尤其在航天航空器、储存罐、船舶、以及传感器中得到广泛使用,如飞机蒙皮、储存罐底、压力仪表中的弹性膜片等。这类结构由于刚度较小,在外界激励下极易产生大振幅的振动,严重影响着整个系统的有效性、服役安全、使用寿命和舒适性等,必须加以研究。然而由于其违背了线性理论的小变形假设,呈现出明显的非线性特征,即几何非线性,导致研究起来非常困难。典型的如圆薄板的大扰度弯曲问题,从基本方程的建立到给出其收敛解中间跨域了近一个世纪。而对于圆薄板的非线性振动问题,尤其是强非线性振动问题,目前依然缺乏非常有效的求解方法。针对圆薄板的非线性振动问题,目前最常使用的是有限元方法。然而在其求解过程中,由于有限元方法无法实现时空完全解耦,即其刚度矩阵显式依赖于时间离散格式。这一方面增大了计算量,因为其刚度矩阵在每一时刻步均需更新。同时,由于时间积分过程中累积的误差,有可能导致结构刚度矩阵存在较大的偏差,进而致使长时间追踪结果失踪,甚至获得错误的近似解。有鉴于此,本课题拟在本小组原有研究的基础之上,探索提出一套分析圆薄板结构非线性行为的高精度小波算法。本文主要内容有:(1)推导了任意平方可积函数在有限区间上(边界Lagrange延拓)基于广义Coiflets小波的逼近公式,对逼近公式在有限区间上的误差给予了证明,并给出了几类在利用小波伽辽金方法求解微分方程的过程中经常遇到的连接系数的推导过程及计算结果;(2)建立了针对中心弹性约束圆薄板大挠度问题的小波求解格式,通过和以往结果对比发现:用多项相乘连接系数离散微分方程所得结果的精度更高;(3)建立了针对圆薄板轴对称非线性振动问题的小波求解格式,并结合Newmark方法对其展开了定量研究,得到了诸如:中心挠度达到板厚2倍时自由振动周期减至线性振动周期65%;薄板中心响应振幅随激励力频率增大而减小等结论。
[Abstract]:Circular thin plates are widely used in various engineering structures, especially in aerospace aircraft, storage tanks, ships and sensors, such as aircraft skin, storage tank bottom, elastic diaphragm in pressure meters, etc. Due to the small stiffness, it is easy to produce large amplitude vibration under external excitation, which seriously affects the effectiveness, service safety, service life and comfort of the whole system, and must be studied. However, due to its violation of the hypothesis of small deformation in linear theory, it presents obvious nonlinear characteristics, that is, geometric nonlinearity, which makes it very difficult to study. Typical problems such as the large perturbed bending of circular thin plates have been used for nearly a century from the establishment of the basic equation to the solution of its convergence. However, for the nonlinear vibration problems of circular thin plates, especially for the strong nonlinear vibration problems, there is still a lack of very effective methods to solve them. Finite element method (FEM) is the most commonly used method for nonlinear vibration of circular thin plates. However, in the process of its solution, the finite element method can not realize the complete decoupling of time and space, that is, its stiffness matrix is explicitly dependent on the time discrete scheme. This increases the computational complexity because the stiffness matrix needs to be updated at every step. At the same time, due to the accumulated errors in the process of time integration, it is possible that there is a large deviation in the stiffness matrix of the structure, which leads to the disappearance of the long time tracking results and even the obtaining of the wrong approximate solution. In view of this, this paper proposes a set of high-precision wavelet algorithms to analyze the nonlinear behavior of circular thin plate structures on the basis of the original research of this group. In this paper, the approximation formula of arbitrary square integrable function on finite interval (boundary Lagrange extension) based on generalized Coiflets wavelet is derived, and the error of approximation formula on finite interval is proved. The derivation process and calculation results of several connection coefficients often encountered in the process of solving differential equations by wavelet Galerkin method are given. A wavelet solution scheme for the large deflection problem of circular thin plates with central elastic constraints is established. By comparing with the previous results, it is found that the accuracy of the results obtained from the discrete differential equations of multiplying the connecting coefficients is higher than that of the previous ones) and a wavelet scheme for solving the axisymmetric nonlinear vibration problems of circular thin plates is established. Based on the Newmark method, some conclusions are obtained, such as the reduction of the free vibration period to the linear vibration period of 65 when the central deflection reaches 2 times the thickness of the plate, and the decrease of the central response amplitude of the thin plate with the increase of the excitation frequency.
【学位授予单位】：兰州大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TB12

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