多自由度参数振动受迫响应的三角级数逼近
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摘要
针对多自由度的参数振动系统,研究了参数周期与激励力周期不相同情况下受迫振动稳态响应的三角级数解。首先根据调制反馈原理将受迫振动响应表示为各谐波成分线性组合的三角级数形式;然后运用谐波平衡及分块带状矩阵求逆算法,求解得到各个谐波项的系数向量;最后将本文提出的级数逼近法与标准的Runge-Kutta法进行了对比。结果表明:两种方法得到的相图结果高度一致,而本文方法在分析和计算上更具有优势:1)用三角级数来表达振动受迫响应,可以直接得到各频率成分的谐波系数,利于参数振动的时域、频域分析;2)所有的谐波系数向量都可以通过计算机进行数值计算,采用该方法得到的逼近误差与级数项数有关,随着级数项数的增加精度会进一步提高;当级数项数取15项时,逼近误差为8.12×10-3;3)对于同样的10万数据点的计算量,Runge-Kutta法耗时69.51s,而本文采用了大型稀疏矩阵快速算法的级数逼近法计算耗时仅0.1317s,计算效率显著提高。
In the paper, we investigate the forced response of multiple freedom parametric vibration system that is governed by a set of ordinary differential equations in which both periodic coefficients and external forces terms have different periods. In this study, the forced response is expressed as a linear combination of harmonic components. By applying harmonic balance, the parametric equations are converted into a set of infinite-order linear algebraic equations. Then by applying algorithm of inverse of block band matrix, all coefficients of the harmonic components in the forced response are fully solved. The accuracy of this approach is verified by comparing resulting phase diagram trajectories with those obtained by the standard Runge-Kutta method. The advantages of the presented approach are:(1) the forced response expressed as a Fourier series is easier to apply in system identification;(2) all coefficients of the harmonic components can be determined by numerical computation;(3) comparing with the standard Runge-Kutta method, it is obviously faster in computing. Thus, the presented approach is suitable for studying the forced response of multiple freedom parametric vibration systems, and it helps to analyze the nonlinear characterization of parametric vibration.
In the paper, we investigate the forced response of multiple freedom parametric vibration system that is governed by a set of ordinary differential equations in which both periodic coefficients and external forces terms have different periods. In this study, the forced response is expressed as a linear combination of harmonic components. By applying harmonic balance, the parametric equations are converted into a set of infinite-order linear algebraic equations. Then by applying algorithm of inverse of block band matrix, all coefficients of the harmonic components in the forced response are fully solved. The accuracy of this approach is verified by comparing resulting phase diagram trajectories with those obtained by the standard Runge-Kutta method. The advantages of the presented approach are:(1) the forced response expressed as a Fourier series is easier to apply in system identification;(2) all coefficients of the harmonic components can be determined by numerical computation;(3) comparing with the standard Runge-Kutta method, it is obviously faster in computing. Thus, the presented approach is suitable for studying the forced response of multiple freedom parametric vibration systems, and it helps to analyze the nonlinear characterization of 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)).
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[10]宫晓春,曹登庆.含参数多自由度非线性系统的降维方法研究[J].动力学与控制学报,2009,7(2):129-135.(Gong Xiaochun,Cao Dengqing.Dimension reduction of nonlinear muti-degreeof-freedom system with multiple parameters[J].Journal of Dynamics and Control,2009,7(2):129-135(in Chinese)).
[11]Dimarogonas A D,Papadopoulos C A.Vibration of cracked shafts in bending[J].Journal of Sound and Vibration,1983,91(4):583-593.
[12]黄迪山.参数振动受迫响应的三角级数解[J].应用数学和力学,2013,34(3):297-305.(Huang Dishan.Trigonometric series approach for forced parametric vibration response[J].Applied Mathematics and Mechanics,2013,34(3):297-305(in Chinese)).
[13]黄迪山,傅晨宸.参数振动受迫响应的三角级数完整解[J].应用力学学报,2013(6):887-893.(Huang Dishan,Fu Chenchen.Complete trigonometric series approach for forced parametric vibration response[J].Chinese Journal of Applied Mechanics,2013(6):887-893(in Chinese)).
[14]Huang Dishan,Fu Chenchen.Forced response approach to predict parametric vibration[J].International Journal of Acoustics and Vibration,2013,18(2):51-57.
[15]黄迪山.复杂参数振动的调制反馈分析[J].应用力学学报,1995,12(2):72-77.(Huang Dishan.Modulating feedback analysis for the intricate parametric vibration[J].Chinese Journal of Applied Mechanics,1995,12(2):72-77(in Chinese)).
[16]Bevilacqua R,Codenotti B,Romani F.Parallel solution of block tridiagonal linear systems[J].Linear Algebra Application,1988,104:39-57.
[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.On computing Floquet transition matrices of rotorcraft[J].Journal of the American Helicopter Society,1981,26(3):56-61.
[4]Sinha S C,Wu D H.An efficient computational scheme for the analysis of periodic systems[J].Journal of Sound and Vibration,1991,151(1):91-117.
[5]David J W,Mitchell L D,Daws J W.Using transfer matrices for parametric system forced response[J].Journal of Vibration,Acoustics,Stress and Reliability in Design,1987,109(6):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):297-308.
[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(7):761-773.
[8]Nayfeh A H,Mook D T.Nonlinear Oscillations[M].New York:Wiley-Interscience,1979.
[9]丁虎,胡超荣,陈立群,等.轴向变速黏弹性Rayleigh梁非线性参数振动稳态响应[J].振动与冲击,2012,31(5):136-138.(Ding Hu,Hu Chaorong,Chen Liqun,et al.Steady state response of nonlinear vibration of an axially accelerating viscoelastic Rayleigh beam[J].Journal of Vibration and Shock,2012,31(5):136-138(in Chinese)).
[10]宫晓春,曹登庆.含参数多自由度非线性系统的降维方法研究[J].动力学与控制学报,2009,7(2):129-135.(Gong Xiaochun,Cao Dengqing.Dimension reduction of nonlinear muti-degreeof-freedom system with multiple parameters[J].Journal of Dynamics and Control,2009,7(2):129-135(in Chinese)).
[11]Dimarogonas A D,Papadopoulos C A.Vibration of cracked shafts in bending[J].Journal of Sound and Vibration,1983,91(4):583-593.
[12]黄迪山.参数振动受迫响应的三角级数解[J].应用数学和力学,2013,34(3):297-305.(Huang Dishan.Trigonometric series approach for forced parametric vibration response[J].Applied Mathematics and Mechanics,2013,34(3):297-305(in Chinese)).
[13]黄迪山,傅晨宸.参数振动受迫响应的三角级数完整解[J].应用力学学报,2013(6):887-893.(Huang Dishan,Fu Chenchen.Complete trigonometric series approach for forced parametric vibration response[J].Chinese Journal of Applied Mechanics,2013(6):887-893(in Chinese)).
[14]Huang Dishan,Fu Chenchen.Forced response approach to predict parametric vibration[J].International Journal of Acoustics and Vibration,2013,18(2):51-57.
[15]黄迪山.复杂参数振动的调制反馈分析[J].应用力学学报,1995,12(2):72-77.(Huang Dishan.Modulating feedback analysis for the intricate parametric vibration[J].Chinese Journal of Applied Mechanics,1995,12(2):72-77(in Chinese)).
[16]Bevilacqua R,Codenotti B,Romani F.Parallel solution of block tridiagonal linear systems[J].Linear Algebra Application,1988,104:39-57.