| 张永康,鲍四元.幂函数形式连续变截面梁振动的弯曲固有频率分析*[J].,2024,43(2):330-338 |
| 幂函数形式连续变截面梁振动的弯曲固有频率分析* |
| Analysis of Natural Frequency of Beams with Power-Function-Type Continuous Variable Section |
| 投稿时间：2023-08-23 修订日期：2024-03-06 |
| 中文摘要: |
| 本文使用微分方程解析法求解变截面梁固有频率。首先，建立变截面梁模型，其中截面面积和惯性矩均按幂次函数变化。得到变截面梁自由振动时挠度的解析表达式，并获得不同边界条件下梁弯曲振动的固有频率方程。其中惯性矩所对应幂指数与截面面积的幂指数的差值为4时，可得自振频率方程的精确形式；而幂指数差值不等于4时，给出近似解法。其次，对4种具体的变截面梁求解不同边界下的自振频率，并与瑞利-里兹法所得的自振频率解比较。验证精确解法结果的正确性，并发现近似解法结果的相对偏差在5%以内。该解析方法较瑞利-里兹法具有能快速求解的特点，且易于分析截面参数对梁固有频率的影响。由算例可得，边界和其他参数不变时，梁的同阶次无量纲自振频率随着幂次指数的增加而增加。几何参数中仅截面形状参数改变时，随着形状参数的增加，梁的同阶次无量纲自振频率随之减小，但固定-自由梁的第一阶自振频率除外。 |
| 英文摘要: |
| In this paper, the natural frequencies of beams with variable section are obtained by using analytical solution method for differential equation. First, a model is established for beam with variable cross-section. The analytical form of the deflection of the beam is derived, and the bending vibration frequency equation is obtained under different boundary conditions. For the cross section of a rod, when the power exponent of the function for moment of inertia is 4 larger than that of the function for area, the exact form of the natural frequency equation is obtained. The approximate form of the natural frequency equation is obtained for the case of the difference of the power exponent being not equal to 4. Then for four kinds of cross-section, the frequencies of the beams are obtained for different boundary conditions. It is found that compared with the results by the Rayleigh-Ritz method the relative derivative is less than 5%, and this proves the accuracy of the presented analytical method. The presented method has the advantage of being a fast-solving method, which can also be used to analyze the influence of the geometric parameters on the natural frequencies. Finally, it can be concluded that the dimensionless natural frequencies of variable section beams increase with the index of the varying cross section if the boundary and other parameters keep unchanged. With the increasing of the shape coefficient, the dimensionless natural frequency of the same order of the beam gradually decreases except for the first order natural frequency of the clamped-free beam. |
| DOI：10.11684/j.issn.1000-310X.2024.02.011 |
| 中文关键词: 变截面梁，固有频率，微分方程解析法 |
| 英文关键词: Beams with variable section, Natural frequency, analytical method for the differential equation |
| 基金项目:国家自然科学基金项目（面上项目，重点项目，重大项目） |
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