受面内激励和横向外激励联合作用下蜂窝夹层板的双Hopf分叉

DOUBLE HOPF BIFURCATIONS OF A HONEYCOMB SANDWICH PLATE SUBJECTED TO TRANSVERAL AND IN-PLANE EXCITATION

随着航空航天事业的发展,对各种材料性能的要求也越来越高.而蜂窝夹层板在结构和性能上具有许多优点,已在航空航天等领域应用广泛,并在一些重要结构中充当承力部件,但由于其特殊的蜂窝结构,相对于一般的板,在受力时会发生比较大的变形,所以用非线性理论研究蜂窝夹层板结构,并考察不同参数对非线性振动特性的影响,具有重要的理论和实际意义.如今,蜂窝夹层板的几何非线性问题已引起更多学者的关注.在一般均质理论的假设下,一些学者已经研究了各向同性蜂窝夹层板板的非线性动力学特性.本文研究了一类受面内激励和横向外激励联合作用下的四边简支蜂窝夹层板在主参数共振-1:2内共振时的双Hopf分叉问题.首先利用多尺度法得到系统的平均方程,然后结合分叉理论得到了系统的分叉响应方程,根据对分叉响应方程的分析,得到了六种不同的分叉响应曲线并给出了系统产生双Hopf分叉的条件.利用数值方法得到系统在参数平面的分叉集,通过对不同分叉区域的分析发现,随着参数的变化系统平衡点会分叉为两类周期解,随后周期解会通过广义静态分叉为准周期解,或者通过广义Hopf分叉为3D环面.

A honeycomb sandwich plate with hexagonal honeycomb core was investigated to reveal the dynamic behavior near a critical point characterized by initial resonance. Based on the averaged equations, the transition boundaries were obtained to divide the parameter space into a set of regions, which correspond to different types of solutions. By applying the stability criteria to determine the stable conditions of respective equilibrium points, the conditions of the occurrence of double Hopf bifurcations were found. Two types of periodic solutions may bifurcate from the initial equilibrium. And the periodic solutions may lose their stabilities via a generalized static bifurcation, which leads to stable quasi-periodic solutions, or via a generalized Hopf bifurcation, which leads to stable 3D tori.