In the present paper, the governing equations of a vibratory beam with moderately large deflection and arbitrary cross-section are derived by using the first-order shear deformation theory. The beam is homogenous, isotropic and it is subjected to the axial loads. The kinematic of the problem is according to the von-Kármán strain-displacement relations and the Hooke law is used as the constitutive equations. The partial differential governing equations describing the axial and transverse vibrations of homogeneous beams contain four coupled nonlinear equations with variable coefficients which are derived employing Hamilton's principle. The Galerkin method in conjunction with the perturbation technique is applied to obtain the linear natural frequencies. A parametric study is performed and the effects of different thickness functions such as linear, polynomial and trigonometric on the results are investigated. The non-linear frequencies which contain the corrections on the linear frequencies are calculated. The corrected parts of the non-linear frequencies are functions of the axial as well as the transverse amplitudes of the vibrations. The influences of the axial load and aspect ratio on the linear and non-linear frequencies are studied too. To confirm the reliability of the vibration analysis carried out in the present paper, the analytical results are checked with the corresponding numerical results obtained from the finite element analysis. The numerical and analytical results are in a good agreement.

中文翻译：

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使用剪切变形理论的变截面梁的线性和非线性振动

本文利用一阶剪切变形理论推导出中等偏转任意截面振动梁的控制方程。梁是均匀的、各向同性的，并承受轴向载荷。该问题的运动学是根据 von-Kármán 应变-位移关系，并使用胡克定律作为本构方程。描述均匀梁轴向和横向振动的偏微分控制方程包含四个具有可变系数的耦合非线性方程，这些方程是采用哈密顿原理推导出来的。Galerkin 方法与微扰技术相结合，用于获得线性固有频率。进行了参数研究，并研究了不同厚度函数（例如线性、多项式和三角函数）对结果的影响。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振幅和横向振幅的函数。还研究了轴向载荷和纵横比对线性和非线性频率的影响。为了确认本文中进行的振动分析的可靠性，分析结果与从有限元分析中获得的相应数值结果进行了核对。数值和解析结果吻合良好。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振幅和横向振幅的函数。还研究了轴向载荷和纵横比对线性和非线性频率的影响。为了确认本文中进行的振动分析的可靠性，分析结果与从有限元分析中获得的相应数值结果进行了核对。数值和解析结果吻合良好。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振幅和横向振幅的函数。还研究了轴向载荷和纵横比对线性和非线性频率的影响。为了确认本文中进行的振动分析的可靠性，分析结果与从有限元分析中获得的相应数值结果进行了核对。数值和解析结果吻合良好。还研究了轴向载荷和纵横比对线性和非线性频率的影响。为了确认本文中进行的振动分析的可靠性，分析结果与从有限元分析中获得的相应数值结果进行了核对。数值和解析结果吻合良好。还研究了轴向载荷和纵横比对线性和非线性频率的影响。为了确认本文中进行的振动分析的可靠性，分析结果与从有限元分析中获得的相应数值结果进行了核对。数值和解析结果吻合良好。