分离变量法求解二维平面薄板相关问题的研究  

Separation of Variables Method for Solving Related Problems of Two Dimensional Planar Thin Plates

作　　者：屈恩相 齐辉[2] 杨杰[3] 王丽 邓琳 乔雪 郭丽婷 Qu Enxiang;Qi Hui;Yang Jie;Wang Li;Deng Lin;Qiao Xue;Guo Liting(School of Architecture and Civil Engineering,Qiqihar University,Qiqihar,Heilongjiang 161006,China;School of Aerospace and Civil Engineering,Harbin Engineering University,Harbin,Heilongjiang 150001,China;School of Mechanical Engineering,Shanghai Dianji University,Shanghai 201306,China)

机构地区：[1]齐齐哈尔大学建筑与土木工程学院,黑龙江齐齐哈尔161006 [2]哈尔滨工程大学航天与建筑工程学院,黑龙江哈尔滨150001 [3]上海电机学院机械学院,上海201306

出　　处：《黑龙江工业学院学报（综合版）》2021年第6期82-89,共8页Journal of Heilongjiang University of Technology(Comprehensive Edition)

基　　金：国家自然科学青年基金项目(项目编号:12002198);黑龙江省教育厅基本科研业务项目(项目编号:135309358);2020年度黑龙江省省属本科高校基本科研业务费面上项目(项目编号:135509142)。

摘　　要：分离变量法作为数学物理方程中常用的方法之一,被广泛的用来解决偏微分方程的定解问题。基本思想是将偏微分方程混合问题经过变量分离,转化为常微分方程的初值问题。将原方程含有的多个变量混合问题拆分成含有若干个只含单一变量的常微分方程。以二维平面薄板为例,采用分离变量法求解该问题的振动方程、波动方程及热传导方程问题的相关研究,推导出平面薄板在给定边界下的解并总结出矩形薄板在不同边界条件下特征值及特征函数的表达。As one of the commonly used methods in mathematical and physical equations,the method of separating variables is widely used to solve the problem of definite solution of partial differential equations.The basic idea is to transform the mixed problem of partial differential equation into the initial value problem of ordinary differential equation through the separation of variables.In fact,the mixed problem of multiple variables contained in the original equation is divided into several ordinary differential equations containing only a single variable.In this paper,taking a two-dimensional thin plate as an example,the vibration equation,wave equation and heat conduction equation of the problem are solved by the method of separating variables.The solutions of planarthin plates under different boundaries are derived and the expressions of eigenvalues and eigenfunctions of rectangular thin plates under given boundary conditions are summarized.

关 键 词：分离变量法 偏微分方程 二维平面薄板 波动方程 热传导方程 

分 类 号：TU435[建筑科学—岩土工程]