Natural Frequencies of Longitudinal Oscillations for a Nonuniform Variable Cross-Section Rod

V. I. Gorbachev

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Natural Frequencies of Longitudinal Oscillations for a Nonuniform Variable Cross-Section Rod

机译：非均匀可变截面杆的纵向振动的固有频率

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The natural frequencies of longitudinal oscillations of a rod such that its Young's modulus, the density, and the cross-sectional area are functions of the longitudinal coordinate are analyzed. For solving the corresponding problem, an integral formula is used to represent the general solution to the original Helmholtz equation with variable coefficients in terms of the general solution to the accompanying equation with constant coefficients. Frequency equations are derived in the form of rapidly converging Leibniz series for three types of boundary conditions. For these cases the frequency zeroth-approximation equations are given to quickly find the lowest natural frequencies with an adequate accuracy.

机译：分析了杆的纵向振动的固有频率，以使其杨氏模量，密度和横截面积成为纵向坐标的函数。为了解决相应的问题，使用积分公式来表示原始的具有可变系数的亥姆霍兹方程的一般解，这是针对具有常数系数的所附方程的一般解。对于三种类型的边界条件，频率方程以快速收敛的莱布尼兹级数的形式导出。对于这些情况，给出了频率为零的近似方程，可以以足够的精度快速找到最低的固有频率。

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《Moscow University Mechanics Bulletin》 |2016年第1期|7-15|共9页
作者
V. I. Gorbachev;
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Moscow State University, Faculty of Mechanics and Mathematics, Leninskie Gory, Moscow, 119899, Russia;

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1. Erratum to: Identification of Cross-Section Defects of the Rod by Using Eigenfrequencies and Features of the Shape of Longitudinal Oscillations (vol 48, pg 561, 2020) [J] . Akulenko L. D., Gavrikov A. A., Nesterov S. V. Mechanics of solids . 2020,第1期

机译：错误：识别杆的横截面缺陷通过使用纵向振动形状的特征（Vol 48，PG 561,2020）
2. Identification of Cross-Section Defects of the Rod by Using Eigenfrequencies and Features of the Shape of Longitudinal Oscillations [J] . Akulenko L. D., Gavrikov A. A., Nesterov S. V. Mechanics of solids . 2019,第8期

机译：利用纵向振动形状的特征和特征识别杆的横截面缺陷
3. LONGITUDINAL VIBRATIONS OF ELASTIC RODS OF VARIABLE CROSS-SECTION [J] . G. M. Ulitin, S. N. Tsarenko International Applied Mechanics . 2015,第1期

机译：截面弹性杆的纵向振动
4. Dynamics of Longitudinal Impact in the Variable Cross-Section Rods [C] . R Stepanov, D Romenskyi, S Tsarenko International Conference on Advanced Engineering and Technology . 2018

机译：可变横截面杆纵向冲击动态
5. A measurement of hadron production cross-sections for the simulation of accelerator neutrino beams and a search for muon neutrino to electron neutrino oscillations in the difference of the squared mass in approximately the 1 squared electron volt region [D] . Schmitz, David W. 2008

机译：测量强子产生横截面，以模拟加速器中微子束，并在大约1平方电子伏特区域的平方质量差中搜索μ中子到电子中微子的振荡
6. Analytical Modeling for the Bending Resonant Frequency of Multilayered Microresonators with Variable Cross-Section [O] . Agustín L. Herrera-May, Luz A. Aguilera-Cortés, Hector Plascencia-Mora, 2011

机译：截面可变的多层微谐振器弯曲共振频率的解析模型
7. Erratum to: Identification of Cross-Section Defects of the Rod by Using Eigenfrequencies and Features of the Shape of Longitudinal Oscillations [O] . L. D. Akulenko, A. A. Gavrikov, S. V. Nesterov 2020

机译：错误：通过使用纵向振荡形状的特征和特征来识别杆的横截面缺陷
8. EFFECTIVE CROSS-SECTIONS AND NEUTRON FLUX DISTRIBUTIONS IN A NATURAL URANIUM ROD APPLICATION TO ROD 683 [R] . S.A. KUSHNERIUK 1957

机译：天然铀棒应用于杆683的有效横截面和中子通量分布

Natural Frequencies of Longitudinal Oscillations for a Nonuniform Variable Cross-Section Rod

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