几何非线性梁振动的渐近分析:基尔霍夫方程和博洛廷方程

Igor V. Andrianov, Steve G. Koblik
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摘要

本文分析了梁的几何非线性振动的各种近似模型。在实践中,简化方程通常基于准静态基尔霍夫假设,忽略了轴向惯性。这一假设的依据是梁在轴向的规定端部位移。在死载荷下,准静态基尔霍夫假说会产生一个线性方程。本文中获得的相应近似方程是基于渐近程序。弯曲刚度与拉伸/压缩刚度的比值被视为一个小参数。第一近似方程中考虑了轴惯性。V.V. Bolotin 提出的 "非线性惯性 "概念得到了讨论。分析了使用准静态基尔霍夫假设时最常见的错误。
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Asymptotic analysis of geometrically nonlinear beam vibrations: Kirchhoff and Bolotin equations
The paper analyzes various approximate models of geometrically nonlinear vibrations of a beam. In practice, simplified equations are often based on the quasi‐static Kirchhoff hypothesis—neglecting axial inertia. This hypothesis is justified with the prescribed end‐displacements of the beam in the axial direction. Under dead loading, quasi‐static Kirchhoff hypothesis results in a linear equation. The corresponding approximate equations obtained in this paper are based on the asymptotic procedure. The ratio of bending stiffness to reduced tensile/compressive stiffness is taken as a small parameter. Axial inertia is taken into account in the equation of the first approximation. Introduced by V.V. Bolotin concept “nonlinear inertia” is discussed. The most common errors in using the quasi‐static Kirchhoff hypothesis are analyzed.
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