A NOTE ON GALERKIN METHOD1)

CHEN Bo, LI Yinghui,2), LI Xingyu

School of Mechanics and Aerospace Engineering, Southwest Jiaotong University, Chengdu, 610031, China

 基金资助: 1)国家自然科学基金资助项目(11872319)

Abstract

Galerkin method is a numerical method widely used in mathematics, physics, and engineering problems. There are some controversies among textbooks and some literature about the selection of trial functions. To make an example, this paper uses Galerkin method to solve the static and dynamic problems of a cantilever beam under the axial load. It is proved that correct results cannot be obtained when choosing trial functions that satisfy only displacement but not force boundary conditions, even if increasing the number of trial functions.

Keywords： Galerkin method; trial function; axial load; cantilever beam

CHEN Bo, LI Yinghui, LI Xingyu. A NOTE ON GALERKIN METHOD1). Mechanics in Engineering, 2022, 44(2): 393-396 DOI:10.6052/1000-0879-21-308

1 轴向力下悬臂梁横向位移和固有频率

$EI\frac{\partial^{4}w}{\partial x^{4}}+P\frac{\partial^{2}w}{\partial x^{2}}+\rho A\frac{\partial^{2}w}{\partial t^{2}}=q\left( {x,t} \right)$

图1

${w}(x,t) \big|_{x=0} =\frac{\partial {w}(x,t)}{\partial x} \Bigg|_{x=0} =0\\ {EI\frac{\partial ^{2}{w}(x,t)}{\partial x^{2}}} \Bigg|_{x=L} =\\ \qquad \left[ {EI\frac{\partial ^{3}{w}(x,t)}{\partial x^{3}}+P\frac{\partial {w}(x,t)}{\partial x}} \right]_{x=L} =0$

1.1 横向位移

$EI\frac{\partial^{4}w_{\rm s} }{\partial x^{4}}+P\frac{\partial^{2}w_{\rm s} }{\partial x^{2}}=q_{\rm s} \left( x \right)$

$w_{\rm s} (x)=\sum\limits_{j=1}^n {a_{j} } \phi_{j} (x)$

$\hspace{-8mm}\phi_{j} (x)=\sin (\beta_{j} x)-\sinh (\beta_{j} x)-\\ \hspace{-8mm}\qquad \frac{\sin (\beta_{j} L)+\sinh (\beta_{j} L)}{\cos (\beta_{j} L)+\cosh (\beta_{j} L)}(\cos (\beta_{j} x)-\cosh (\beta_{j} x))$

${\widetilde{{K}}}_{n\times n} {\widetilde{{A}}}_{n\times 1} ={Q}_{n\times 1}$

$\left.\begin{array}{l}\widetilde{K}_{i j}=\int_{0}^{L}\left(E I \phi_{i}^{\prime \prime \prime \prime}+P \phi_{i}^{\prime \prime}\right) \phi_{j} \mathrm{~d} x \\\widetilde{A}_{i 1}=a_{i}, \quad Q_{i 1}=\int_{0}^{L} q \phi_{i} \mathrm{~d} x\end{array}\right\}$

$\widetilde{A}_{n\times 1} =\widetilde{K}_{n\times n}^{-1}{Q}_{n\times 1}$

1.2 固有频率

$EIW""+PW"-\rho A\omega^{2}W=0$

$W(x)=\sum\limits_{j=1}^n {{b}_{j} } \phi_{j} (x)$

$\left( {{\hat{{K}}}_{n\times n} -{\hat{{M}}}_{n\times n} \omega^{2}} \right){\hat{{A}}}_{n\times 1} ={\bf0}$

$\left.\begin{array}{rl}\hat{K}_{i j} & =\int_{0}^{L}\left(E I \phi_{i}^{\prime \prime \prime \prime}+P \phi_{i}^{\prime \prime}\right) \phi_{j} \mathrm{~d} x \\\hat{M}_{i j} & =\rho A \int_{0}^{L} \phi_{i} \phi_{j} \mathrm{~d} x, \quad \hat{A}_{i 1}=b_{i}\end{array}\right\}$

$\left| {{\hat{{K}}}_{n\times n} -{\hat{{M}}}_{n\times n} \omega^{2}} \right|=0$

2 验证和讨论

$\left.\begin{array}{c}\xi=\frac{x}{L}, \quad \bar{w}_{\mathrm{s}}=\frac{w_{\mathrm{s}}}{L}, \bar{P}=\frac{P}{P_{\mathrm{cr}}} \\\bar{q}_{\mathrm{s}}=\frac{q_{\mathrm{s}} L^{3}}{E I}, \quad \bar{\omega}=\omega \sqrt{\frac{\rho A L^{4}}{E I}}\end{array}\right\}$

参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Han F, Dan D, Cheng W.

Exact dynamic characteristic analysis of a double-beam system interconnected by a viscoelastic layer

Composites Part B$:$ Engineering, 2019, 163:272-281

Li XY, Wang XH, Chen YY, et al.

Bending, buckling and free vibration of an axially loaded timoshenko beam with transition parameter: direction of axial force

Int J Mech Sci., 2020, 176:105545

/

 〈 〉