Abstract

This article presents an analytical procedure to predict the beams nonlinear natural frequencies using the first-order shear deformation theory. The nonlinear kinematics assumptions include the moderately large deformation of the mid-plane stretching and transverse deflection that are defined by using the von Kármán relations. The strains are small and Hooke’s law is used as the constitutive equation. A coupled nonlinear longitudinal-transverse set of motion equations are derived utilizing Hamilton’s principle. The equations are solved analytically using the powerful multiple scales method of the perturbation technique. At first, the linear natural frequencies and mode shapes are determined. Then the nonlinear frequencies which contain the corrections on the linear frequencies are calculated. The corrected parts of the nonlinear frequencies are functions of the axial and transverse amplitudes of vibrations. The influences of the axial load and aspect ratio on the linear and nonlinear frequencies are studied too. The results of the special cases are compared with the available results in the literature and the finite element analysis. The results show the noticeable effect of the axial amplitude as well as the transverse amplitude of vibrations on the nonlinear frequencies.

摘要

本文介绍了使用一阶剪切变形理论预测梁非线性固有频率的分析过程。非线性运动学假设包括使用 von Kármán 关系定义的中平面拉伸和横向偏转的适度大变形。应变很小，使用胡克定律作为本构方程。利用哈密顿原理推导了一组耦合的非线性纵向-横向运动方程。方程是解析求解的使用微扰技术的强大的多尺度方法。首先，确定线性固有频率和振型。然后计算包含对线性频率的校正的非线性频率。非线性频率的校正部分是振动的轴向和横向振幅的函数。还研究了轴向载荷和纵横比对线性和非线性频率的影响。将特殊情况的结果与文献和有限元分析中的可用结果进行比较。结果表明振动的轴向振幅和横向振动振幅对非线性频率有显着影响。