Euler杆大挠度屈曲解析逼近解的构造

Construction of Analytical Approximate Solutions to Buckling of Euler's Column with Large Deflection

基于Euler杆大挠度屈曲的控制方程,构造了屈曲载荷及最大挠度的高精度解析逼近解.利用Maclaurin展开和Chebyshev多项式将控制方程中的正弦项用三次多项式近似代替,得到一个Duffing型方程,再将牛顿法与谐波平衡法相结合解对应的Duffing方程,从而给出Euler杆大挠度屈曲的解析逼近解.求解过程中只需解线性方程组即可构造出屈曲载荷及最大挠度的解析逼近公式.几乎在自变量的全部取值范围内,给出的公式都有较高的逼近精度.

Based on the approximate solutions to governing equation the buckling load of buckling of the Euler＇ s column with large deflection, analytical and the largest deflection of the column have been established First, the sine term that appears in the governing equation is replaced with a polynomial of degree 3 by means of the Maclaurin series expansion and the Chebyshev polynomial approximation. Subsequently, the resulting Duffing equation is approximately solved by combing the Newton＇ s method with the method of harmonic balance. The method yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution. The new analytical approximations for the buckling load and the largest deflection of the Euler＇ s column show an excellent agreement with the numerically exact ones, and are valid over nearly the whole range of the independent variable.