柔性约束下压杆的一些稳定和不稳定的临界状态

Some Stable and Unstable Critical States of a Compression Rod With a Flexible Support

研究了一端固定、一端弹簧约束滑动固定的压杆在Euler临界载荷作用下的稳定性.将系统的势能表示为转角的泛函,将扰动量展开成Fourier级数,将势能的二阶变分表示成一个二次型,得到在临界状态下势能的二阶变分半正定,并求得临界载荷与屈曲模态.进一步研究临界状态下高阶变分的正定性,包括四阶和六阶变分的正定性.结果表明,与刚性约束不同的是,柔性约束压杆临界状态的稳定性与约束的刚度有关,有稳定与不稳定之分,并给出了临界状态是稳定和不稳定的情况下柔性约束相对刚度的范围.

Under Euler＇ s critical load, the stability of a slender compression rod with one end fixed and the other clamped in rotation but translationally restrained by a spring was studied. The potential energy of the system was expressed with the functional of the rod deflection angle ; the disturbance was expanded into the Fourier series ; the 2nd-order variation of the potential energy was expressed with a quadratic form. The 2nd-order positive semidefinite variation in the critical state was derived with the buckling mode and the critical load obtained. A further study of the positive definiteness of higher-order variations, including the 4th and 6th varia- tions, indicates that the stability of the compression rod with a flexible support is related to the stiffness of the flexible constraint and may be stable or unstable, which is different from the case of a rigid constraint. In the stable and unstable critical states the ranges for the relative stiffness of the flexible support were also given.