非线性振动系统周期解的数值分析
A Numerical Treatment of the Periodic Solutions of Non-Linear Vibration Systems
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摘要: 用直接数值积分法求非线性振动系统的周期解,求解时对初始条件进行迭代,使它与终点条件相一致.积分时间区间(即周期)或运动方程中的某些参数,也可在迭代过程中随同变化,积分方法是变步长的. 用这种“打靶”法求周期解,所需计算工作量相对较少.其中误差主要来源于数值积分,故不难估计并控制它足够小.这种方法可处理各种类型的振动问题,如单自由度和多自由度系统的自由无阻尼振动、强迫振动、自激振动和参数振动等等;也能求得不稳定解和那些对参数变动十分敏感的解.解的稳定性根据相关的周期系数微分方程来研究.求共振曲线或其他振动特性曲线时,利用插值方法并自动调节步长来定出迭代始值. 为了阐明这种方法的通用性,计算了若干例子.非线性的描述可用解析函数或任何其他形式,例如分段线性函数.文中还就所得周期解指出了非线性振动的一些值得注意的性质.部分计算结果与已有的近似解或实验结果作了比较.Abstract: Direct numerical integration can be used to find the periodic solutions for the equations of motion of nonlinear vibration systems. The initial conditions are iterated so that they coincide with the terminal conditions. The time interval of the integration (i.e.,the period) and certain parameters of the equations of motion can be included in the iterations.The integration method has a variable steplength.This shooting method can produce periodic solutions with a shorter computer time. The only error occurs in the numerical integration and it can therefore be estimated and made small enough. Using this method one can treat a variety of vibration problems, such as free conservative, forced, parameter-excited and self-sustained vibrations with one or several degrees-of-freedom. Unstable solutions and those which are sensitive to parameter changes can also be calculated.The stability of the solutions is investigated based on the theory of differential equations with periodic coefficients. The extrapolation method and the procedure of automatic steplength control are used to estimate the initial values of iterations by determining the resonance curve and other vibration characteristics.Some examples have been calculated to illustrate the applicability of the method. The non-linearity way be expressed by an analytical function or any other functions, such as a piecewise linear function. Several remarkable features of nonlinear vibrations are presented through the periodic solutions obtained. Finally, some results are compared with those obtained by other approximation methods and experiments.
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[1] Hsu,C.S.,Limit cycle oscillations of parametrically excited second-order nonlinear systems,J.Appl.Mech.,42(1975),176-182. [2] Riganti,R.,A Study on the Forced Vibrations of a Class on Nonlinear Sgstems with Application to the Duffing Equation Part II:Numerical Treatment,Mechanica,11(1976),81-88. [3] Mayfeh,A.H.and D.T.Mook,Nonlinear Oscillations.John Wiley & Sons,New York-Chichester-Brisbane-Toronto(1979). [4] Poincaré,H.,Mémoire sur les courbes définies par une équations différentielles,J.Math.3,Série,7(1881),375-422. [5] Hsu,C.S.,Nonlinear Behaviour of Multibody Systems under Impulsive Parametric Excitation,in"Dynamics of Multibody Systems,"Springer,Berlin-Heidelberg-New York(1977). [6] Hsu,C.S.,On Nonlinear Parametric Excitation Problems,Adv.Appl.Mech.,17(1977),245-301. [7] Urabe,M.,Numerical determination of periodic solution of nonlinear sgstem,J.Sci.Hiroshima Univ.Ser.A,20(1957),125-148. [8] Urabe,M.,Infinitesimal deformation of cycles.J.Sci.Hiroshima Univ.Ser.A.18(1954),37-53. [9] Urabe,M.,Remarks on periodic solutions of Van der Pol's equation,J.Sci.Hiroshima Univ.Ser.A,24(1960),197-199. [10] Urabe,M.,Nonlinear Autonomous Oscillations,Akademic Press,New York-London(1967). [11] Ruf.W.-D.,Numerische Lösung des Diffing-Problems.Diplomarbeit,Institut A fur Mechanik,Uni.Stuttgart(1978). [12] Ling,F.H(凌复华).,Numerische Bereahung periodischer Lösungen einiger nichtlinearer Schwingungssysteme,Dissertation,Uni.Stuttgart(1981). [13] Poincare,H.,Les Méthodesnouvelles de la mecanique céleste Vol.1,Guathiervillars,Paris(1892). [14] Ляпунов,А.М.,Обмая Задача об устойчивости Движения,Харвков,(1982),или ОНТИ(1935) [15] 马尔金,《非线性振动理论中的李维普诺夫方法与邦加来法》,科学出版社,(1959) [16] Малкин,И.Г.,Теорая Устойчивости Движения,Гостехиздат,(1952) [17] Kane,T.R.and D.Sobala,A new method for attitude stabilization,AIAA J.,1(1963),1365-1367. [18] Stoer,J.and R.Bulirsch,Einführung in die Numerische Mathematik Ⅱ,Springer,Berlin-Heidelberg-New York(1978). [19] Fehlberg,E.,Klassische Runge-Kutta Formeln funfter und siebenter Ordnung mit Schrittweiten-Kontrolle,Computing,4(1969),93-106. [20] Fehlberg,E.,Klassische Runge-Kutta Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Warmeleitungsprobleme,Computing,6(1970),61-71. [21] Shampine,L.F.and M.K.Gordon,Computer Solution of Ordinary Differential Equations,The initial Value Problem,W.H.Freema and Company,San Francisco(1975). [22] Stoker,J.J.,Nonlinear Vibrations in Mechanical and Electrical Systems,Interscience Publishers,New York-London(1950). [23] Schrapel,H.D.,Erweiterung eines Satzes von Andronow und Witt,ZAMM 57(1977),T89-T90. [24] Moler,C.and C.Van Loan,Nineteen dubious ways to compute the exponential of a matrix,SIAM Rev.,20(1978),801-836. [25] Friedmann,P.,C.E.Hammond and T.-H.Woo,Efficient numerical treatment of periodic systems with application to stability problems,Int.J.NUm.Math.Eng.,11(1977),1117-1136. [26] Hsu,C.S.,Impulsive parametrix excitation:theory,J.Appl Mech.,39(1972),551-558. [27] Hsu,C.S.and W.H.Cheng,Applications of the theory of impulsive parametric excitation and new treatments of general parametrix excitation problems,J.Appl.Mech.,40(1973),78-86. [28] Urabe,M.and A.Reiter,Numerical computation of nonlinear forced oscillations by Galerkin's procedure,J.Math.Anal.Appl.14(1966),107-140. [29] Rosenberg,R.M.and C.P.Atkinson,On the natural modes and their stability in nonlinear two-degree-of-freedom systems,J.Appl.Mech.,26(1959),377-385. [30] Sehtna,P.R.,Steady-state undamped vibrations of a class of nonlinear discrete systems,J.Appl Mech.,27(1960),187-195. [31] Van Dooren,R.,Differential tones in a damped mechanical system with quadratic and cubic non-linearities,Int.J.Nonlinear Mech.,8. (1973),575-583.
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