研究轴向运动导电梁在平行导线产生的磁场环境中的主-内联合共振问题。基于电磁场基本理论和哈密顿原理，导出轴向运动梁在外激励和磁场共同作用下的非线性振动方程。针对一端夹支一端铰支的导电梁，采用多尺度法求解方程，得到非线性方程的近似解析解和幅频响应方程，并对稳态解的稳定性进行了分析。通过算例，得到系统前两阶幅值随频率调谐参数、外激励力、轴向速度、电流强度等参数的变化规律。结果表明，系统发生主-内联合共振时一阶和二阶响应都被激发，且存在不同的多解区域；一阶和二阶幅值的稳态解个数在几个多解区域同步变化，其个数取决于外激励力、运动速度和电流强度值。
Abstract
The principle-internal resonance of an axially moving conductive beam in the magnetic field induced by parallel wires is investigated. Based on the theory of electromagnetic field and Hamilton principle, the nonlinear vibration equation of the beam under external excitation and magnetic field is derived. For a conductive beam with one side clamped and the other side hinged, the approximate analytical solution and the amplitude frequency response equations for the nonlinear equation are derived by the multiple-scale method, and the stability of the steady-state solutions are also analyzed. Through numerical examples, the corresponding amplitude curves of the first two order vibration modes varying with different frequency tuning parameters, external excitation force, axial velocity and current intensity are obtained. The results show that the first and the second-order response are both excited, and different multi-solution regions are found. The number of steady-state solutions of the first and second-order amplitudes changes simultaneously in the multi-solution regions, and the number depends on the external excitation force, moving velocity and current intensity value.
关键词
磁弹性 /
导电梁 /
轴向运动 /
主-内联合共振 /
多尺度法 /
伽辽金法
{{custom_keyword}} /
Key words
magneto-elastic /
conductive beam /
axially moving /
principal- internal resonance /
multiple-scale method /
Galerkin method
{{custom_keyword}} /
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] 陈贵清, 董保珠, 邱家俊. 水电机组两相线间短路时的参、强联合共振研究[J]. 浙江大学学报: 工学版, 2012, 46(7): 1207-1212.
CHEN Guiqing, DONG Baozhu, QIU Jiajun. Combined parametric and forced resonance of hydro-generator under short circuit between two phases[J]. Journal of Zhejiang University: Engineering Science, 2012, 46(7): 1207-1212.
[2] 杨志安, 李自强. 电机轴承转子多频激励系统参—强联合共振[J]. 机械强度, 2013, 32(5): 695-699.
YANG Zhian, LI Ziqiang. Parametric and forced resonance of the bearing rotor multifrequencies excitation of a motor[J]. Journal of Mechanical Strength, 2013, 35(5): 695-699.
[3] 姜 源, 申永军, 温少芳. 分数阶van der Pol振子的超谐与亚谐联合共振[J]. 振动工程学报, 2019, 32(5): 863-873.
JIANG Yuan, SHEN Yongjun, WEN Shaofen. Super-
harmonic and sub-harmonic simultaneous resonance of fractional-order van der Pol oscillator[J]. Journal of Vibration Engineering. 2019, 32(5): 863-873.
[4] 李 航, 申永军, 李向红, 等. Duffing 系统的主-亚谐联合共振[J]. 力学学报, 2020, 52(2): 514-521.
LI Hang, SHEN Yongjun, LI Xianghong, et al. Primary and subharmonic simultaneous resonance of duffing oscillator[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(2): 514-521.
[5] 魏明海, 肖仪清. 内外共振联合作用下索－梁组合结构非线性振动分析[J]. 振动与冲击, 2012, 31(7): 79-84.
WEI Minghai, XIAO Yiqing. Nonlinear vibration analysis for a cable-beam coupled system under simultaneous internal and external resonances[J]. Journal of Vibration and Shock, 2012, 31(7): 79-84 .
[6] 唐有绮, 陈立群. 面内平动黏弹性板非线性振动的内-外联合共振[J]. 应用数学和力学, 2013, 34(5): 480-487.
TANG Youqi, CHEN Liqun. Internal-external combination resonance of nonlinear vibration of in-plane translating viscoelastic plates[J]. Applied Mathematics and Mechanics, 2013, 34(5): 480-487.
[7] TANG Y Q, ZHANG D B, GAO J M. Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions[J]. Nonlinear Dynamics, 2016, 83(1-2): 401-418.
[8] 冯志华, 胡海岩. 直线运动柔性梁非线性动力学—主参数共振与内共振联合激励[J]. 振动工程学报, 2004, 17(2): 16-131.
FENG Zhihua, HU Haiyan. Nonlinear dynamics of flexible beams undergoing a large linear motion of basement: principal parametric and internal resonances[J]. Journal of Vibration Engineering[J], 2004, 17(2): 16-131.
[9] FENG Z H, HU H Y. Principal parametric and three-to-one internal resonances of flexible beams undergoing a large linear motion[J]. Acta Mechanica Sinica, 2003, 19(4): 354-364.
[10] 冯志华, 胡海岩. 直线运动柔性梁非线性动力学—组合参数共振与内共振联合激励[J]. 振动工程学报, 2004, 17(3): 253-257.
FENG Zhihua, HU Haiyan. Nonlinear dynamics of flexible beams undergoing a large linear motion of basement: combinational parametric and internal resonances[J]. Journal of Vibration Engineering, 2004, 17(3): 253-257.
[11] 黄建亮, 陈树辉. 轴向运动体系的横向非线性振动的联合共振[J]. 振动工程学报, 2005, 18(1): 19-23.
HUANG Jianliang, CHEN Shuhui. Combination resonance of laterally nonlinear vibration of axially moving systems[J]. Journal of Vibration Engineering, 2005, 18(1): 19-23.
[12] 黄建亮, 陈树辉. 外激励力作用下的轴向运动梁非线性振动的联合共振[J]. 振动工程学报, 2011, 24(5): 455-460.
HUANG Jianliang, CHEN Shuhui. Combination resonance of nonlinear forced vibration of an axially moving beam[J]. Journal of Vibration Engineering, 2011, 24(5): 455-460.
[13] HU Y D, WANG J. Principal-internal resonance of an axially moving current-carrying beam in magnetic field[J]. Nonlinear Dynamics, 2017, 90(1): 683–695.
[14] 胡宇达, 张立保. 轴向运动导电导磁梁的磁弹性振动方程[J]. 应用数学和力学, 2015, 36(1): 70-77.
HU Yuda, ZHANG Libao. Magneto-elastic vibration equations for axially moving conductive and magnetic beams[J]. Applied Mathematics and Mechanics, 2015, 36( 1): 70-77.
[15] 张金志. 轴向运动矩形薄板的磁弹性非线性振动[D]. 秦皇岛: 燕山大学, 2013, 5.
ZHANG Jinzhi. Magneto-elastic nonlinear vibration of axially moving rectangular thin plate [D]. Qinghuangdao: Yanshan University, 2013, 5.
[16] HU Y D, Hu P, Zhang J Z. Strongly nonlinear subharmonic resonance and chaotic motion of axially moving thin plate in magnetic field[J]. Journal of Computational and Nonlinear Dynamics, 2015, 10(2): 021010.
[17] 胡宇达, 戎艳天. 磁场中轴向变速运动载流梁的参强联合共振[J]. 中国机械工程, 2016, 27(23): 3197-3207.
HU Yuda, RONG Yantian. Combined parametric and forced resonance of axially accelerating and current-carrying beam under magnetic field[J]. China Mechanical Engineering,
2016, 27(23): 3197-3207.
{{custom_fnGroup.title_cn}}
脚注
{{custom_fn.content}}
PDF(3416 KB)
