Characteristics analysis and parameter design of a bending vibrational conical horn based on two kinds of beam theories

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Journal of Vibration and Shock ›› 2024, Vol. 43 ›› Issue (2) : 114-122.

CHANG Tingting，SHEN Feng，BAO Siyuan

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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.

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Timoshenko beam theory / horn / a new type of improved Fourier series / free vibration / natural frequency

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CHANG Tingting，SHEN Feng，BAO Siyuan. Characteristics analysis and parameter design of a bending vibrational conical horn based on two kinds of beam theories[J]. Journal of Vibration and Shock, 2024, 43(2): 114-122

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[1]张然,贺西平,王照伟.具有大振幅放大系数的倒锥形变幅杆[J].振动与冲击,2020,39(24):143-149. ZHANG Ran,HE Xiping,WANG Zhaowei. An inverted cone horn with large amplitude amplification factor. JOURNAL OF VIBRATION AND SHOCK, 2020, 39(24): 143-149(in Chinese). [2]张雄,焦锋. 超声加工技术的应用及其发展趋势[J]. 工具技术,2012,01:3-8. Zhang Xiong, Jiao Feng. Applications and Development Trends of Ultrasonic Machining Technology[J]. Tool Technology,2012,01:3-8(in Chinese). [3] 万志坚,梅红.一种面内弯纵复合模态的直线超声电动机研究[J].微特电机,2021,49(04):17-21,29. Wan Zhi-jian, Mei Hong. Research on a linear Ultrasonic Motor using bending-in-plane and longitudinal modes. Small & Special Electrical Machines, 2021,49(04),17-21,29(in Chinese). [4] 刘向尧,聂宏,魏小辉.多跨的三种梁的横向自由振动模型[J].振动与冲击,2016,35(08):21-26. LIU Xiang-yao NIE Hong WEI Xiao-hui . The transverse free vibration model of three multi-span beams. Journal of vibration and shock, 2016, 35(8): 21-26(in Chinese). [5] 牛国华,王刚锋,王剑.组合L型变截面梁的自由振动特性分析[J].振动与冲击,2022,41(05):228-234. Niu Guo-hua, Wang Gangfeng, WANG Jian. Analysis of free vibration characteristics of composite L-shaped variable cross-section beam. Journal of vibration and shock, 2022, 41(5): 228-234(in Chinese). [6] Ma H. Exact solution of vibration problems of frame structures[J]. International Journal for Numerical Methods in Biomedical Engineering, 2010, 26(5):587-596. [7] Wang Xinwei, Yongliang Wang，Yong Zhou. Application of a new differential quadrature element method to free vibrational analysis of beams and frame structures[J]. Journal of Sound and Vibration, 2004, 269(3-5):1133-1141. [8] 吴林潮,吴志渊,张文明.基于近场动力学的梁结构振动特性分析方法研究[J].固体力学学报,2021,42(04):467 -475. Linchao Wu, Zhiyuan Wu, Wenming Zhang. Vibration analysis method of beam structure in Peridynamics[J]. Chinese Journal of Solid Mechanics, 2021, 42(4): 467-475(in Chinese). [9]HAN R P S, ZU J W. Analytical dynamics of a spinning Timoshenko beam subjected to a moving load[J].Journal of the Franklin Institute, 1993, 330(1):113-129. [10] Hamioud Saida, Khalfallah Salah. Spectral element analysis of free - vibration of Timoshenko beam[J]. International Journal for Engineering Modelling, 2018, 31(1/2):61-76. [11] 李晓姣. 梁力学问题的辛本征值分析方法[D].大连理工大学,2018. (Li Xiaojiao.Symplectic eigenvalue direction method for beam mechanics problems[D]. Dalin University of Technology, 2018 (in Chinese)). [12] Mohammad Karkon. An efficient finite element formulation for bending, free vibration and stability analysis of Timoshenko beams[J]. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2018, 40(10)):1-16. [13]严日明，刘德福，陈涛，等. 一种圆锥形变幅杆弯曲振动固有频率的计算方法[J]. 振动与冲击, 2016, 35(7): 198-204. YAN Ri-ming, LIU De-fu, CHEN Tao, etc. A calculation method of flexural vibration frequency for conical horn. JOURNAL OF VIBRATION AND SHOCK, 2016, 35(7): 198-204(in Chinese). [14] 葛仁余,张金轮,韩有民,等.功能梯度变截面梁自由振动和稳定性研究[J].应用力学学报, 2017,34(05):875-880, 1012. Ge Renyu, Zhang Jinlun, Han Youmin , etc. Free vibration and stability of axially functionally graded beams with variable cross-section. CHINESE JOURNAL OF APPLIED MECHANICS, 2017,34(05):875-880, 1012(in Chinese). [15] Auciello N M . Free vibration of a restrained shear-deformable tapered beam with a tip mass at its free end[ J]. Journal of Sound and Vibration, 2000, 237: 542-549. [16] Auciello N M, Ercolano A.A general solution for dynamic response of axially loaded non-uniform Timoshenko beams[ J] .International Journal of Solids and Structures, 2004, 41:4861-4874. [17] Li W L. Free vibrations of beams with general boundary conditions[J]. Journal of Sound and Vibration. 2000,237(4):709-725. [18] LI W L, ZHANG X, DU J, et al. An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports[J]. Journal of Sound and Vibration, 2009, 321(1-2): 254-269. [19]吴子奇,王治,史冬岩,等. 任意边界条件下变厚度中厚矩形板的静动态特性分析[J]. 船舶力学,2018,22(11):1407-1420. WU Zi-qi;WANG Zhi;SHI Dong-yan, etc. Static and dynamic characteristics analysis of moderately thick rectangular plates with arbitrary boundary supports.Journal of Ship Mechanics. 2018,11(22): 1407-1420(in Chinese). [20] Bao S, Wang S, Wang B. An improved Fourier–Ritz method for analyzing in-plane free vibration of sectorial plates[J]. Journal of Applied Mechanics, 2017, 84(9): 091001. [21] 蒋济同,王熙堃. 一种推导Timoshenko 梁频率方程的新方法[J]. 中国海洋大学学报(自然科学版),2008(06): 1035-1039. Jiang Ji-tong, Wang Xi-kun. A new method for deriving frequency equations of Timoshenko beam[J], Periodical of Ocean University of China(Natural sciences),2008(06): 1035-1039(in Chinese). [22] 胡海岩.梁在固有振动中的对偶关系[J].力学学报,2020,52(01):139-149. Hu Haiyan. Duality relations of beams in natural vibrations. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(1): 139-149(in Chinese).