兰州大学机构库 >土木工程与力学学院
小波数值方法及其在薄板结构非线性分析中的应用
Alternative TitleWavelet numerical method and its application in nonlinear analysis of thin plate structures
段岳华
Thesis Advisor周又和
2017-05-10
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name硕士
Keyword圆薄板大挠度 非线性振动 广义Coiflets小波 数值计算 微分方程 微分积分方程
Abstract

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。

Other Abstract

Circular thin plates are widely used in various engineering structures, especially in the circular parts of aerospace, storage tanks and sensors, such as aircraft skin, storage tank bottom, elastic diaphragm in pressure instrument, etc. Because of the small stiffness of them, the vibration of large amplitude is easy to occur under the external excitation, which seriously affects the effectiveness of the whole system, service safety, service life and comfort, and must be studied. However, because of its violation of the small deformation hypothesis of linear theory, it presents obvious nonlinear characteristics, namely geometric nonlinearity, which makes it very difficult to study. The large deflection bending problem of thin plate, which is a typical example, from the establishment of the basic equation to getting the convergence solution, spanned nearly a century. For the nonlinear vibration of circular thin plate, especially the strongly nonlinear vibration problem, there is still lack of very effective method.

For the nonlinear vibration of circular thin plate, the numerical method most commonly used is finite element method (FEM). However, in the process of its solution, the finite element stiffness matrix is explicitly dependent on the discrete-time format. This increases the amount of calculation, because the stiffness matrix needs to be updated at each step. At the same time, due to the accumulated error in the process of time integration, it may lead to large deviation of the structural stiffness matrix, resulting in the long time tracking results disappeared, and even the wrong approximate solution.

In view of this, this paper intends to explore a set of high precision wavelet algorithm to analyze the nonlinear behavior of circular thin plate. The main contents of this paper contain: (1) The approximation formula of arbitrary L2 function on a finite interval (Lagrange extension) based on generalized Coiflets wavelet is deduced, the quantitative error analysis is given, and the derivation process and calculation results of several kinds of connection coefficients often encountered in the process of solving differential equations by Wavelet-Galerkin method are given. (2) A wavelet solution scheme for solving the large deflection problem of the center elastic constraint plate is established, and compared with the previous results, it is found that the accuracy of the results obtained by the multiple multiplication connection coefficients is higher. (3) The wavelet solution format for solving the axisymmetric nonlinear vibration problem is established, and the quantitative research is carried out in combination with the Newmark method, some conclusions are obtained, such as: the free vibration period is reduced to 65% of the linear vibration cycle when the center deflection reaches the thickness of plate;The response amplitude of the center of circular thin plate decreases with the increase of the exciting force frequency.

URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/225850
Collection土木工程与力学学院
Recommended Citation
GB/T 7714
段岳华. 小波数值方法及其在薄板结构非线性分析中的应用[D]. 兰州. 兰州大学,2017.
Files in This Item:
There are no files associated with this item.
Related Services
Recommend this item
Bookmark
Usage statistics
Export to Endnote
Altmetrics Score
Google Scholar
Similar articles in Google Scholar
[段岳华]'s Articles
Baidu academic
Similar articles in Baidu academic
[段岳华]'s Articles
Bing Scholar
Similar articles in Bing Scholar
[段岳华]'s Articles
Terms of Use
No data!
Social Bookmark/Share
No comment.
Items in the repository are protected by copyright, with all rights reserved, unless otherwise indicated.