摘 要

随着科技的发展，人类活动范围的扩大，使得各种大型高精尖的结构和工具不断更新，也意味着所面临的工作环境更加恶劣，各类工程领域对材料的要求更加复杂。圆形薄板在航空航天、船舶工程等领域有巨大的利用价值，本文研究了爆炸载荷作用下弹性固定圆形薄板的大挠度动力响应。

基于薄板的大挠度振动理论推导了极坐标下圆形薄板非线性振动的控制偏微分方程和应变协调方程。假设包含两个待定常数的圆形薄板的横向位移函数，根据弹性固定圆形薄板的面外边界条件确定两个待定常数，从而得到了满足弹性固定边界条件的圆形薄板的横向位移的表达式；根据应变协调方程和圆形薄板的面内边界条件确定了Airy应力函数的表达式。将圆形薄板的横向位移的表达式和Airy应力函数的表达式代入圆形弹性薄板非线性振动的控制偏微分方程，利用Galerkin法推导了弹性固定圆形薄板的二阶非线性运动常微分方程。利用MATLAB求解二阶非线性运动常微分方程即可得到弹性固定圆形薄板中心的横向位移的时间历程。采用Friedlander方程描述简化的爆炸载荷，利用ANSYS WORKBENCH有限元软件计算了爆炸载荷作用下圆形薄板的几何非线性动力响应，研究了圆形弹性薄板的几何参数、弹性固定程度以及爆炸载荷的参数对其非线性动力响应的影响，并与由迦辽金法得到的相应结果进行了比较。

关键词：圆形薄板；爆炸载荷；大挠度；Galerkin法；有限元法

Abstract

With the development of science and technology and the expansion of the range of human activities, all kinds of large and sophisticated structures and tools are constantly updated, which also means that the working environment is worse and the requirements for materials in various engineering fields are more complex. Circular plates have great value in aerospace, ship engineering and so on. Thus, it is valuable to study the large-deflection dynamic response of circular plates, particularly, when it suffer from an explosion load.

Based on the theory of large-deflection vibrations of thin plates, the partial differential equations and strain coordination equations for nonlinear vibration of circular thin plates in polar coordinates are derived. Assuming the lateral displacement function of a circular thin plate containing two undetermined constants, the two undetermined constants are determined according to the out-of-plane boundary conditions of the elastically fixed circular thin plate, thereby obtaining the expression of the lateral displacement of the circular thin plate satisfying the elastically fixed boundary conditions ; The expression of the Airy stress function is determined according to the strain coordination equation and the in-plane boundary conditions of the circular thin plate. Substituting the expressions of the lateral displacement of the circular thin plate and the expressions of the Airy stress function into the partial differential equations for the nonlinear vibration control of the circular elastic thin plate, the second-order nonlinear ordinary differential equations for the elastically fixed circular thin plate are derived using the Galerkin method . Using MATLAB to solve the second-order nonlinear ordinary differential equation of motion can obtain the time course of the lateral displacement of the center of the elastic fixed circular thin plate. Using the Friedlander equation to describe the simplified explosive load, the ANSYS WORKBENCH finite element software was used to calculate the geometric nonlinear dynamic response of the circular thin plate under the explosive load. The effects of its nonlinear dynamic response are compared with the corresponding results obtained by the Galerkin method.

Key words: circular elastic thin plate; explosive load; large deflection; Galerkin method; finite element method

目录

摘要 I

Abstract II

第1章 绪论 1

1.1研究背景及意义 1

1.2国内外研究现状 1

1.4 主要研究内容和技术路线 2

1.4.1 主要研究内容 2

1.4.2技术路线 3

第2章 圆形薄板非线性振动的控制运动偏微分方程的推导 4

2.1研究对象 4

2.2 研究所采用的假设 5

2.3 运动微分方程的推导 5

第3章 圆形薄板非线性振动的常微分方程的推导 12

3.1边界条件 12

3.2圆形薄板的横向位移函数 12

3.3 圆形薄板非线性振动的常微分方程 13

第4章 爆炸载荷作用下弹性固定的圆形薄板的非性动力响应的数值计算 16

4.1参数设置 16

4.2载荷模型 16

4.2.1爆炸简介 16

4.2.2爆炸波压力 17

4.2.3均匀的爆炸载荷模型 18

4.3 MATLAB数值求解 19

4.3.1 MATLAB基本信息 19

4.3.2数值计算 19

4.3.3 计算结果 19

4.3.4不同参数对圆板非线性动力响应的影响 21

4.4小结 25

第5章 爆炸载荷作用下圆形弹性薄板的非线性动力响应的有限元分析 26

5.1 ANSYS软件和有限元分析 26

5.1.1 ANSYS软件的简单介绍 26

5.1.2结构动力学分析 26

5.2 有限元模型的建立 27

5.2.1计算类型 27

5.2.2材料创建 28

5.2.3几何建模 28

5.2.4网格划分以及边界条件 29

5.2.4载荷加载 29

5.3 本章小结 29

第6章 有限元数值计算结果与分析 30

6.1圆形薄板的整体变形 30

6.2圆形弹性薄板中心的横向位移 31

6.2.1 几何参数的影响 31

6.2.2弹性固定程度的影响 33

6.2.3载荷参数的影响 34

第7章 总结与展望 38

7.1 主要结果 38

7.2展望 39

致谢 40

参考文献 41

附录 43

A1 微分方程常系数计算C文件 43

A2 MATLAB数值计算程序文件 45