研究变速旋转圆板的非线性磁弹性参强联合振动问题。给出旋转圆板在磁场中的磁弹性振动方程，应用伽辽金法离散变量，得到横向磁场中旋转圆板轴对称参强联合振动微分方程。运用多尺度法求解振动微分方程，分析久期项得到系统发生参强联合共振时的两种共振状态，并分别给出两种状态下系统的幅频响应方程。通过数值计算，给出圆板的协调参数、磁场、转速、激励力等参数变化对振动特性的影响，对比两种共振条件下的幅值-参数曲线，讨论不同参数变化对系统稳定性的影响。通过系统的全局分岔图，讨论分岔参数变化对系统动力学特性的影响。
Abstract
Magneto-elastic resonance of a conductive circular plate rotating with varying velocity under combined parametric and forced excitations was investigated. The conductive circular plate was subjected to parametric excitations due to the time-varying rotating speed and magnetic field forces. The magneto-elastic parametric vibration equations of the variable-velocity rotating conductive circular plate were established, its axisymmetric vibration differential equation under combined parametric and forced excitations was obtained through the application of Galerkin method. Then, the multi-scale method was applied to derive two conditions for resonance occurring, two corresponding amplitude-frequency response equations were deduced, respectively. The influences of plate’s coordination parameters, magnetic field parameters, rotating speed and excitations on the vibration performance of the circular plate were studied. Amplitude-parameter curves of two resonance conditions were compared, and the influences of parameters on the system’s stability were discussed. According to the global bifurcation diagram of the system, the influences of changes of bifurcation parameters on the system dynamic characteristics were discussed.
关键词
磁弹性 /
圆板 /
参强联合共振 /
旋转运动 /
多尺度法 /
稳定性
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Key words
magneto-elastic /
circular plate /
resonance under combined parametric and forced excitations /
rotary motion /
multi-scale method /
stability
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参考文献
[1] Bekhoucha F, Rechak S, Duigou L, et al. Nonlinear forced vibrations of rotating composite beams with internal combination resonance [J]. Design and Modeling of Mechanical Systems, 2013, Part II: 159 - 166.
[2] Pratiher B, Dwivedy S K. Nonlinear vibrations and frequency response analysis of a cantilever beam under periodically varying magnetic field [J]. Mechanics Based Design of Structures and Machines, 2011, 39(3): 378 - 391.
[3] 胡宇达, 徐耀玲, 白象忠. 横向磁场中圆板的轴对称振 [J]. 振动与冲击, 1998, 17(4): 71 - 74.
HU Yu-da, XU Yao-ling, BAI Xiang-zhong. The axial symmetric vibration of circular plate in transverse magnetic field [J]. Journal of Vibration and Shock, 1998, 17(4): 71 - 74.
[4] 胡宇达. 轴向运动导电薄板磁弹性耦合动力学理论模型 [J]. 固体力学学报, 2013, 34(4): 417 - 425.
HU Yu-da. Magneto-elastic coupled dynamics theoretical model of axially moving current-conducting thin plate [J]. Chinese Journal of Solid Mechanics, 2013, 34(4): 417 - 425.
[5] 胡宇达, 杜国君. 磁场环境下导电圆形薄板的磁弹性强迫振动[J]. 工程力学, 2007, 24(7): 184 - 188.
HU Yu-da, DU Guo-jun. Forced vibration of a thin round conductive plate in magnetic field [J]. Engineering Mechanics, 2007, 24(7): 184 - 188.
[6] 李哲, 胡宇达, 姚臻臻. 旋转运动导电圆板磁弹性振动分析[J]. 燕山大学学报, 2015, 39(5): 464 - 470.
LI Zhe, HU Yu-da, YAO Zhen-zhen. Forced oscillation of conductive rotating circular plate in magnetic field [J]. Journal of Yanshan University, 2015, 39(5): 464 - 470.
[7] Hu Yuda, Hu Peng, Zhang Jinzhi. 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 - (1 - 12).
[8] Özhan B B, Pakdemirli M. Principal parametric resonances of a general continuous system with cubic nonlinearities [J]. Applied Mathematics and Computation. 2012, 219(5): 2412 - 2423.
[9] Shahgholi M, Khadem S E. Primary and parametric resonances of asymmetrical rotating shafts with stretching nonlinearity [J]. Mechanism and Machine Theory. 2012, 51: 131 - 144. [10] 胡宇达, 孙建涛, 张金志. 横向磁场中轴向变速运动矩形板的参数振动 [J]. 工程力学, 2013, 30(9): 299 - 304.
HU Yu-da, SUN Jian-tao, ZHANG Jin-zhi. Parametric vibration of axially accelerating rectangular plate in transverse magnetic field [J]. Engineering Mechanics, 2013, 30(9): 299 - 304.
[11] Hu Yu-da, Zhang Jin-zhi. Principal parametric resonance of axially accelerating rectangular thin plate in magnetic field [J]. Applied Mathematics and Mechanics, 2013, 34(11): 1405 - 1420.
[12] 张伟, 霍拳忠. 参数激励与强迫激励联合作用下非线性振动系统的分叉 [J]. 力学学报, 1991, 23(4): 464 - 474.
ZHANG Wei, HUO Quan-zhong. Bifurcations of nonlinear oscillation system under combined parametric and forcing excitation [J]. ACTA Mechanica Sinica, 1991, 23(4): 464 - 474.
[13] 康厚军, 赵跃宇, 蒋丽忠. 参数振动和强迫振动激励下超长拉索的面内非线性振动 [J]. 中南大学学报(自然科学版), 2011, 42(8): 2439 - 2445.
KANG Hou-jun, ZHAO Yue-yu, JIANG Li-zhong. In-plane nonlinear vibration of super long stay cables under parametric and forced excitations [J]. Journal of Central South University, 2011, 42(8): 2439 - 2445.
[14] 贾启芬, 于雯, 邱家俊, 等. 机电非线性振动系统参、强联合激励的分岔 [J]. 机械强度, 2003, 25(6): 624 - 627.
JIA Qi-fen, YU Wen, QIU Jia-jun, et al. Bifurcation on nonlinear by the combined parametric and forced resonances in an electromechanical vibration system [J]. Journal of Mechanical Strength, 2003, 25(6): 624 - 627.
[15] 刘延柱, 陈立群. 非线性振动 [M]．北京: 高等教育出版社, 2001.
LIU Yan-zhu, CHEN Li-qun. Nonlinear vibrations [M]. Beijing: Higher Education Press, 2001.
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