二自由度参数振动自由响应逼近
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摘要
对二自由度参数振动问题,应用反馈调制概念,将参数振动的自由振动响应表示成为以振荡频率和参数激励频率的线性组合,用矩阵三角级数进行逼近。应用谐波平衡,将二自由度参数振动方程转化为无限阶线性代数方程组;从齐次方程非零解得到特征方程,通过数值解得到主振荡频率;引入归一化模态,求解出模态、系数矩阵及自由响应通解,由初始条件确定自由响应的任意常数;定义一个计算误差函数,将该方法和龙格-库塔法进行比较,当逼近级数项数大于一定项时,计算误差值比龙格-库塔法要小的多。因此,所给的矩阵三角级数表达为二自由度参数振动自由响应逼近提供了一种有效的解析分析工具,它具有理论研究和工程应用的价值。
For a 2-DOF parametric vibration system, using the concept of feedback modulation, its free response was expressed as a linear combination of vibrations with its natural frequencies and parametric excitation frequencies, and approximated with matrix trigonometric series. Adopting the harmonic balance method, the 2-DOF parametric vibration equation was converted into an infinite set of linear algebraic equations, the characteristic equation was obtained from the nonzero solution to homogeneous linear algebraic equations, and the system's main natural frequencies were achieved from numerical solutions to the characteristic equation. Introducing normalized modes, the system's modal coefficient matrix and the general solution of its free response were solved. With initial conditions, arbitrary constants of its free response's general solution were determined. A computation error function was defined and used to compare the proposed approach and Runge-Kutta algorithm. The computation results showed that when terms of the approximation series are larger than a certain number, the former's computational error is much smaller than the latter's one; the proposed method provides an effective analytical tool for free response approximation of a 2-DOF parametric system, and is valuable for theoretical study and engineering application.
For a 2-DOF parametric vibration system, using the concept of feedback modulation, its free response was expressed as a linear combination of vibrations with its natural frequencies and parametric excitation frequencies, and approximated with matrix trigonometric series. Adopting the harmonic balance method, the 2-DOF parametric vibration equation was converted into an infinite set of linear algebraic equations, the characteristic equation was obtained from the nonzero solution to homogeneous linear algebraic equations, and the system's main natural frequencies were achieved from numerical solutions to the characteristic equation. Introducing normalized modes, the system's modal coefficient matrix and the general solution of its free response were solved. With initial conditions, arbitrary constants of its free response's general solution were determined. A computation error function was defined and used to compare the proposed approach and Runge-Kutta algorithm. The computation results showed that when terms of the approximation series are larger than a certain number, the former's computational error is much smaller than the latter's one; the proposed method provides an effective analytical tool for free response approximation of a 2-DOF parametric system, and is valuable for theoretical study and engineering application.
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.
[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 D H,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,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(4):356-360.
[6] DELTOMBES R,MORAUX D,PLESSIS G,et al.Forced response of structural dynamic systems with local time-dependent stiffnesses[J].Journal of Sound and Vibration,2000,237(5):761-773.
[7] WU W T,WICKERT J A,GRIFFIN 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.
[8] HUANG J L,CHEN S H.Combination resonance of nonlinear forced vibration of an axially moving beam[J].Journal of Vibration Engineering,2011,24(5):455-460.
[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] HAN Q,WANG J,LI Q.Experimental study on dynamic characteristics of linear parametrically excited system[J].Mechanical Systems and Signal Processing,2011,25(5):1585-1597.
[11] HAN Q,WANG J,LI Q.Parametric instability of a cantilever beam subjected to two electromagnetic excitations:Experiments and analytical validation[J].Journal of Sound and Vibration,2011,330(14):3473-3487.
[12] CHEN C C,YEH M K.Parametric instability of a beam under electromagnetic excitation[J].Journal of Sound and Vibration,2001,240(4):747-764.
[13] HUANG D.Forced response approach of a parametric vibration with a trigonometric series[J].Mechanical Systems and Signal Processing,2015,52:495-505.
[14] HUANG D,ZHANG Y,SHAO H.Free response approach in a parametric system[J].Mechanical Systems and Signal Processing,2017,91:313-325.
[15] SINHA S C,REDKAR S,DESHMUKH V,et al.Order reduction of parametrically excited nonlinear systems:techniques and applications[J].Nonlinear Dynamics,2005,41(1):237-273.