【摘要】 <正> 一.前言 由于技术的发展,结构的动力稳定问题得到了愈来愈广泛的注意。最早研究这类问题的是别遼也夫,他处理了铰支杆件受周期性变化的纵向力的动力稳定问题,他把这个问题化为马休(Mathieu)型的方程,因而比较容易地确定了稳定的条件。至于薄壁杆件的动力稳定问题,则是由果耳金布拉特在1947年提出,这个问题在一般的场合是相当困难的,他研究了截面具有一个或二个对称轴的铰支杆件受作用于弯曲中心的纵向周期力的情形。如果截面具有二对称轴,则基本方程化为三个独立的马休-希尔(Hill)型方程,因而比较容易建立稳定的条件。如果截面具有一个对称轴,则基本问题更多还原

【Abstract】 In this paper, an approximate method for determining the unstable regions of dynamic stability of thin-walled beams is given. The beam is assumed to be under the actions of concentrated longitudinal forces at both ends and of the typewhere P₀ =const., P₁（t） a periodic force with period 2π/（ω） and μ a small parameter. The end conditions are arbitrary. By using trigonometric series or Galerkin’s method satisfying the end conditions, the fundamental equations, based on Vlasof’s theory, are reduced to a system of three ordinary linear differential equations （4） or （7） of 2^nd order with periodic coefficients. Moreover, they can easily be transformed to canonical formTherefore, their characteristic equations are reciprocal equations, with characteristic roots symmetrically distributed with respect to the real axis and unit circle in a complex plane. The condition of boundary lines between stable and unstable regions is taken as, that all of the characteristic roots have unit modulus （absolute value）, but there exist equal roots. Expanding the characteristic exponentials in series of the small parameter μ, this condition is represented by the following equations:where ω_niω_nk represent different frequencies of n-mode vibrations of the beam under the action of a constant force P₀, and ω is the frequency of P₁（t）. When UUUU-0, （27） and （28） becomeHence, dynamic unstability would take place at the neighbourhoods of these critical ratios, expressed by （29） and （30）.When the unstable regions of dynamic stability are desired, we use perturbation method to determine a_njⁱ. For practical use, it is sufficient to determine a_nj¹, a_nj² only. The boundary lines can then be approximately determined by the following equations:For illustrating this method, a simply supported beam of narrow rectangular cross-section, under the action of varying end moments （fig. 2） is considered. The fundamental unstable regions, corresponding to bending, torsional and "mixed" type of dynamic unstability are calculated and shown in figs. 4, 3, 5.更多还原