四边简支加劲板的几何非线性自由振动及内共振

Geometrically nonlinear free vibration and internal resonance of a stiffened plate with four edges simply supported

研究了加劲板非线性振动的求解方法与振动特性。将加劲板分为母板与加劲肋两个部分考虑,其中母板视为大挠度板,加劲肋视为Euler梁。分别建立板与加劲肋的应变能与动能的表达式,并用张量的形式表示。将应变能与动能代入Lagrange方程,得到一系列关于面内与面外广义坐标的非线性振动微分方程组,多模态解可以通过增量迭代法求出,而单模态解可以用椭圆函数表示。最后,对一个四边简支且不可移动的加劲板前4阶模态进行分析,分别讨论两个方向设置不同数量加劲肋的情况下非线性自振频率与振幅的关系,并分析系统的内共振,得到了加劲板非线性振动一些特性,对工程设计有一定的参考意义。

One approach was presented to investigate the nonlinear free vibration characteristics of a stiffened plate. The stiffened plate was divided into a plate and stiffeners. The plate was considered to be geometrically nonlinear, and the stiffeners are taken as Euler beams. The strain and kinetic energy equations of both the plate and stiffeners were formulated, they were expressed with tensors. Substituting the strain and kinetic energy equations into Lagrange equation, a series of nonlinear differential equations with respect to in-plane and out-plane generalized coordinates were obtained. The multi-mode solutions were obtained with the incremental-iterative method. The exact single-mode solution could be given in terms of elliptic functions. At last, a stiffened plate with four immovable simply supported edges was studied. Selecting the first four modes, the relationship between the corresponding nonlinear natural frequency and its amplitude was discussed with the number of stiffeners in two directions varying. Besides, the internal resonance of the system was studied. Some nonlinear vibration characteristics of the stiffened plate were obtained, they provided a reference for engineering design.