1∶2内共振情况下点阵夹芯板动力学的奇异性分析

Abstract

Inner resonance is a typical nonlinear dynamic behavior, and the symmetric crossply composite sandwich plates have been widely used in aerospace. The studies about inner resonance of such sandwich plates have both theoretical and engineering significances. Based on the dynamic equations for the sandwich-plates, of which the boundary conditions were simply supported on 4 sides, the transverse and inplane excitations were both considered. The average equations in the polar form were obtained with the multiscale method, and the algebraic equations in the steady state form were derived through the average equations. The singularity theory was utilized to investigate 1∶2 resonant bifurcations of the symmetric crossply sandwich plates. Based on the algebraic equations in the steady state form, the restricted tangent space was obtained for the bifurcation equations with 2 tuning parameters, an inplane excitation and a transverse excitation. Then the algebraic equations were simplified under strong equivalence, and the normal form of the algebraic equations were obtained in nondegenerate cases. The singularity theory were generalized for the general nonlinear dynamic equations with 2 state variables and 4 bifurcation parameters, and the 18 universal unfoldings of bifurcation equations with codimension 4 were obtained in the case of 1∶2 internal resonance. The transition sets in the parameter plane and the bifurcation diagrams were depicted. The relationships between the tuning parameters and the exciting parameters were determined when bifurcation, hysteresis, and double limit points happened. The numerical results indicate that the vibration modes in different bifurcation regions are different.