基于Chebyshev-变分法的复杂开口形状矩形薄板弯曲振动特性分析

针对复杂开口形状的矩形薄板弯曲振动问题,提出一种基于Chebyshev-变分原理的建模方法,建立弹性边界条件下不同开口形状矩形薄板弯曲振动模型。采用边界约束因子模拟弹性边界条件,视开口部分为一种物理属性为零的特殊薄膜。将板的横向位移展开成双重Chebyshev多项式级数形式,建立薄板的拉格朗日泛函,利用变分法推导薄板的特征方程并求得固有频率及对应振型。开展开口薄板模态试验研究,对比理论计算结果与试验结果及有限元结果,验证该方法及模型的准确性和有效性。研究边界约束和开口形状对弯曲振动特性的影响。结果表明:开口形状对结构低阶固有频率影响较小,对高阶固有频率影响较大;开口形状的改变对结构奇数阶固有频率的影响大于对偶数阶固有频率的影响。

Bending vibration characteristics analysis of a rectangular plate with complex-opening using Chebyshev-variational method

A modeling method for the vibration analysis of a rectangular plate with different openings under elastically restrained conditions was proposed based on the Chebyshev-variational theory.Its elastic boundary condition was simulated by introducing boundary constraint factors.The opening portion was regarded as a membrane with zero physical properties.The transverse vibration displacement was expressed in a double Chebyshev series form, based on which the characteristic equation of the plate with opening was obtained by using the variational method.The modal tests of the plate with different openings were carried out.By comparing the experimental results with the corresponding theoretical results and finite element results, the accuracy of the present method was validated.The effects of boundary constraint and opening shape on bending vibration characteristics were analyzed.It is shown that the shape of the opening has little effect on the low-order natural frequencies of the structure, but has a great influence on the high-order natural frequencies.Furthermore, the shape of the opening shows greater influence on odd-order natural frequencies than that on even-order natural frequencies.