基于两种梁理论对变幅锥形杆弯曲振动的特性分析及参数设计

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摘要为了研究圆锥形杆自由振动的特性，分别基于欧拉-贝努力梁理论和铁木辛柯梁理论，建立变截面杆自由振动的分析模型。首先采用一种含三角函数的级数形式来表示欧拉-贝努利梁理论下杆的位移函数，以满足端部位移的条件；然后利用能量泛函极小化得到系数满足的线性方程组，进而获得不同边界条件下圆锥形杆在欧拉-贝努利梁理论下的若干阶固有频率；类似地，假设位移的级数形式并利用能量函数，建立锥形杆基于铁木辛柯梁理论的求解方法，可得各阶固有频率和模态；最后给出等截面杆在两种理论下固有频率的转化公式，并推广应用到圆锥形杆的固有频率近似转化。算例分析锥形杆截面参数对结构固有频率的影响，并基于目标设计频率和若干限制条件对锥形杆的尺寸进行设计。数值结果表明，在应用欧拉-贝努利梁理论和铁木辛柯梁理论时，所提方法都能够稳定收敛且计算效率高，具有较高的精确度。本文的研究工作为超声工程中变幅杆的动力学特性提供了计算依据。

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	常婷婷
	沈峰
	鲍四元

关键词 ：铁木辛柯梁理论, 变幅杆, 新型改进傅里叶级数, 自由振动, 固有频率

Abstract：To investigate the free vibration characteristics of conical rod, analysis models are established based on Euler-Bernoulli beam theory and Timoshenko beam theory respectively. First, in the presented models,series forms including trigonometric basis functions are adopted to express the the displacement function, which can satisfy the boundary condition of the rod. Then the linear equation set about the unknown coefficients is obtained based on the the minimum principle of the energy functional, and several natural frequencies and modes can be worked out by the eigenvalue problems. Similarly, suppose the series forms of the displacement and use the energy function, the solution method is established for conical rod’s vibration problem based on Timoshenko beam theory, and natural frequencies and modes can be obtained. Finally, a transformation formula is presented to transform the natural frequency under the Euler-Bernoulli beam theory and Timoshenko beam theory. The transformation formula is extended to the case of conical rod. The numerical examples analyze the sectional parameters’ influence on the natural frequency of conical rod, and some length parameters of the conical rod are designed based on value of target natural frequency and some restricted conditions. The numerical results show that under the two classical beam theories the presented method is stably converged with high efficiency and good accuracy. The research can provide computational basis for the dynamic characteristics of horn in ultrasonic engineering.

Key words： Timoshenko beam theory horn a new type of improved Fourier series free vibration natural frequency

收稿日期: 2023-01-03 出版日期: 2024-01-28

引用本文:

常婷婷,沈峰,鲍四元. 基于两种梁理论对变幅锥形杆弯曲振动的特性分析及参数设计[J]. 振动与冲击, 2024, 43(2): 114-122.
CHANG Tingting，SHEN Feng，BAO Siyuan. Characteristics analysis and parameter design of a bending vibrational conical horn based on two kinds of beam theories. JOURNAL OF VIBRATION AND SHOCK, 2024, 43(2): 114-122.

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http://jvs.sjtu.edu.cn/CN/ 或 http://jvs.sjtu.edu.cn/CN/Y2024/V43/I2/114

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