参数振动受迫响应的三角级数完整解
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摘要
应用调制反馈模型给出了一个受迫参数系统响应解的完整数学表达,其响应解可表示为谐波线性组合的三角级数。通过谐波平衡、极限运算方法将响应中谐波分量的所有系数由级数表示,从而给出了一个完整的受迫响应表达,并构建了理论频谱。此外,还定量地确定了参数变量β与主共振频率左偏的关系、图示频率响应中因参数变量β引起的相位跃变等重要响应性质;对受迫响应的相图与标准的龙格-库塔法的结果进行了轨迹一致性比较。研究结果表明:参数振动响应解的三角级数表达在拓宽受迫响应和非线性动力学特性方面具有表达形式简单、分析便捷的优势,可为涉及受迫参数振动理论的研究和工程应用提供参考。
The forecasting model on modulation feedback is used to investigate a forced response of linear system that is governed by an ordinary differential equation with periodic coefficient. The system is excited by both periodic coefficient and external force with the same period. In the developed approach, the forced response is expressed as a linear combination of the harmonic terms. By the application of harmonic balance and limitation operation, all coefficients of harmonic components in the response solution fully approach to a set of series. This approach has been successfully verified by the comparison with the standard Runge-Kutta method. Meanwhile, some important response characteristics, such as the parameter index β effect on shift of main resonance frequency, the parametric index β causing additional phase delay in the frequency response, etc., are discussed through mathematical deduction and computation. Results obtained reveal that the developed approach has an advantage in the complete and analytical solution of forced response and nonlinear dynamic characteristics, and it is very significant for the theoretical research and engineering application concerning the problem of forced parametric vibration.
The forecasting model on modulation feedback is used to investigate a forced response of linear system that is governed by an ordinary differential equation with periodic coefficient. The system is excited by both periodic coefficient and external force with the same period. In the developed approach, the forced response is expressed as a linear combination of the harmonic terms. By the application of harmonic balance and limitation operation, all coefficients of harmonic components in the response solution fully approach to a set of series. This approach has been successfully verified by the comparison with the standard Runge-Kutta method. Meanwhile, some important response characteristics, such as the parameter index β effect on shift of main resonance frequency, the parametric index β causing additional phase delay in the frequency response, etc., are discussed through mathematical deduction and computation. Results obtained reveal that the developed approach has an advantage in the complete and analytical solution of forced response and nonlinear dynamic characteristics, and it is very significant for the theoretical research and engineering application concerning the problem of forced parametric vibration.
引文
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[2]胡海岩.应用非线性动力学[M].北京:航空工业出版社,2000.(Hu Haiyan.Application of nonlinear dynamic[M].Beijing:Aviation Industry Press,2000.(in Chinese))
[3]Gaonkar G H,Simha Prasad D S,Sastry S.Oncoming Floquet transition matrices of rotorcraft[J].Journal of the American Helicopter Society,1981,26(3):56-61.
[4]Sinha S C,Wu Der-Ho,Juneja V,et al.Analysis of dynamic systems with periodically varying parameters via Chebyshev polynomials[J].Journal of Vibration and Acoustics,1993,115(1):96-102.
[5]David J W,Mithchell L D.Using transfer matrices for Parametric system forced response[J].Journal of Vibration,Acoustics,Stress and Reliability in Design,1987,109(4):356-360.
[6]Wu W T,Wickert J A,Griffin J H.Modal analysis of the steady state response of a driven periodic linear system[J].Journal of Sound and Vibration,1995,183(2):186-297.
[7]Deltombe R,Moraux D,Plessis G,et al.Forced response of structural dynamic systems with local time-dependent stiffness[J].Journal of Sound and Vibration,2000,237(5):761-773.
[8]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.(in Chinese))
[9]Dimarogonas A D,Papadopoulos C A.Vibration of cracked shafts in bending[J].Journal of Sound and Vibration,1983,91(4):583-593.
[10]王建军,韩勤锴,李其汉.参数振动系统频响特性研究[J].振动与冲击,2010,29(3):103-108.(Wang Jianjun,Han Qinkai,Li Qihan.Frequency response characteristics of parametric vibration system[J].Journal of Vibration and Shock,2010,29(3):103-108.(in Chinese))
[11]黄迪山,程耀东,童忠钫.参数振动的调制反馈分析[J].浙江大学学报:自然科学版,1992(S1):16-24.(Huang Dishan,Cheng Yaodong,Tong Zhongfang.Modulating feedback analysis for the parametric vibration[J].Journal of Zhejiang University:Natural Science,1992(S1):16-24.(in Chinese))