This paper deals with an analysis of the free vibration of nonprismatic thin-walled beams, with a special focus on the effect of a change in the support point location on the eigenfrequencies of the systems. A change in the support point location is understood here as occurring within the same fixed cross-section of the beam where the latter is supported. The original elements of this study are a thin-walled beam model and a method of solving differential equations, not previously used by other authors. The equations describing the model used in this paper were derived using the momentless theory of shells and the Vlasov theory assumptions. The displacement equations were derived relative to an arbitrary rectilinear reference axis. In most works known to the authors the equations describing the vibration of the nonprismatic thin-walled beam do not take into account the effects due to the curvature of the axis formed by the shear centers. The recursive algorithm was used to solve the obtained differential equations with variable coefficients. The algorithm enables one to solve the analyzed displacement equations in the form of series relative to the orthogonal Gegenbauer polynomials. For special values of the parameter defining the order of the polynomials the Gegenbauer polynomials became Chebyshev or Lagrange polynomials. In the provided numerical example, the effect of a change in the support point location within the fixed cross-section is examined. It is also analyzed which of the approximation polynomials (Chebyshev or Lagrange polynomials) yield more precise results for a small approximation base. In order to verify the model and the effectiveness of the adopted solution method, the results obtained using this method are compared with the results yielded by FEM. The results obtained showed a significant effect of the position of the support point, within a fixed section, on the eigenfrequencies values of the thin-walled systems.

中文翻译：

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支撑点位置变化对薄壁梁振动的影响分析

本文对非棱柱形薄壁梁的自由振动进行了分析，特别关注支撑点位置的变化对系统特征频率的影响。支撑点位置的变化在这里被理解为发生在支撑梁的相同固定横截面内。这项研究的原始元素是薄壁梁模型和求解微分方程的方法，其他作者以前没有使用过。本文使用的描述模型的方程是使用壳的无矩理论和 Vlasov 理论假设得出的。位移方程是相对于任意直线参考轴导出的。在作者已知的大多数作品中，描述非棱柱薄壁梁振动的方程没有考虑由于剪切中心形成的轴曲率而产生的影响。采用递推算法求解得到的变系数微分方程。该算法使人们能够以相对于正交格根鲍尔多项式的级数形式求解所分析的位移方程。对于定义多项式阶数的参数的特殊值，格根鲍尔多项式变为切比雪夫或拉格朗日多项式。在提供的数值示例中，检查了固定横截面内支撑点位置变化的影响。还分析了哪些近似多项式（切比雪夫或拉格朗日多项式）对于小的近似基会产生更精确的结果。为了验证模型和所采用求解方法的有效性，将使用该方法得到的结果与有限元法得到的结果进行了比较。获得的结果表明，在固定截面内，支撑点的位置对薄壁系统的特征频率值有显着影响。