【作者】 李鹏松；

【导师】 吴柏生；

【作者基本信息】 吉林大学 ， 计算数学， 2004， 博士

【摘要】 物理和力学中的振动系统通常用非线性微分方程来描述。对于某些特殊情况，用线性化的微分方程代替非线性微分方程，能够给出原非线性方程的一些有用结果。但多数情况下，这种线性化是不合理的，此时，只能直接研究非线性微分方程。数学上求解线性微分方程的通用理论和方法已经非常成熟，但是，对于任意的非线性微分方程的通用性质却知之甚少。一般情况下，对于非线性方程的研究仅限于一些特殊的方程，而且，为了获得某个非线性微分方程的解析逼近解，求解方法通常要涉及到有限的解析逼近技术中的一种或几种。在求解非线性微分方程的所有解析逼近方法中，应用最广泛研究也最多是小参数摄动方法。这些方法将非线性微分方程的解展开为小参数的级数，主要包括：L-P 法、KBM 法和多尺度法。但是，这些方法只能用于求解弱非线性振动问题。谐波平衡法是另一类求解非线性微分方程的解析逼近方法，该方法用截断的 Fourier 级数逼近非线性微分方程的解。它的突出特点是不要求非线性微分方程中非线性项是小量，即不要求方程中含有小参数。谐波平衡法一般要求非线性恢复力? f (x)为 x 的奇函数( x 代表任意瞬时运动质点到稳定平衡点的距离)， 否则该方法在求非线性振动方程的低阶解析逼近解时会导出矛盾的情况。此外，应用谐波平衡法及其改进方法构造非线性振 1<WP=112>摘 要动的高阶解析逼近解是很困难的，因为这要求复杂的非线性代数方程组的解析解。 本文针对一维保守系统大振幅非线性振动问题，提出了几种新的解析逼近方法。提出的这些新方法不要求非线性振动方程中含有小参数及位移的线性项。这些方法的主要优点是求解过程简单，建立的解析逼近周期与周期解既适用于小振幅又适用于大振幅，特别也包括振幅趋于无穷的情形，且都能给出精度非常高的解析逼近结果。1. 求解大振幅非线性振动问题的修正谐波平衡法 考虑如下的一维保守系统非线性振动方程 d2x + f (x)= 0, (1a) dt2 x(0) = β , dx (0)= 0. (1b) dt 引入新变量τ = ωt ，方程(1a,b)可以写为 ω2x′′+ f (x)= 0, (2a) x(0) = β , x′(0) = 0 (2b)式中()表示对τ 求导。 新变量τ 的选取使得方程(2a,b)的解是关于τ 的以2π ’为周期的周期函数，相应的原非线性振动的周期为T = 2π ω ，周期解 x(τ)及频率ω 都与振幅β 有关。 令 x(τ)= x0(τ)+ ?x0(τ), 其中 x0(τ)是满足初始条件的周期解 x(τ)的初始逼近，?x0(τ)是待求的周期解的校正部分，它们都是关于τ 的以2π 为周期的周期函数。将非线性方程(2a,b)在 x(τ)= x0(τ)关于增量?x0(τ)线性化得 2<WP=113>摘 要 ω2x0′′ + f (x0)+ω2?x0′′ + fx(x0)?x0 = 0, (3a) ?x0(0)= 0, ?x0′(0)= 0. (3b)用谐波平衡法求解关于?x0(τ)的线性方程(3a,b)可以给出原非线性振动方程的解析逼近周期与周期解。 当非线性恢复力? f (x)为 x的奇函数 [ f (? x)= ? f (x)]时，令 x0(τ ) = β cosτ , (4a) ?x0(τ)= ci{cos[(2i ?1)τ]? cos[(2i +1)τ]}. ∑N (4b) i=1 当非线性恢复力? f (x)为 x的一般函数 [ f (? x)≠ ? f (x)]时，令 x0(τ ) =β +α β ?α + cosτ , (5a) 2 2 ?x0(τ)= ci{cos(iτ)?cos[(i + 2)τ]}. ∑N (5b) i=0 此外，我们还将修正谐波平衡法用于 Duffing-harmonic 振子以及非自然系统振动问题，通过例子说明了此时本文提出的方法仍然能够给出高精度解析逼近周期与周期解。2. 求解大振幅非线性振动问题的迭代法 引入新变量τ = ωt，将方程(1a,b)写为 x′′ + x = x ?1 ω2 f (x):= g(x). (6a) x(0) = β , x′(0) = 0 (6b)式中()表示对τ 求导。新变量τ 的选取使得方程(6a,b)的解是关于τ 的以2π ’ 3<WP=114>摘 要为周期的周期函数。相应的原非线性振动的周期为T = 2π ω ，周期解 x(τ)及频率ω 都与振幅β 有关。 提出如下的迭代过程 xk′′+1 + xk = g(xk )+ gx(xk )(xk ? xk ), +1 ?1 ?1 ?1 (7a) xk (0) = β, xk′+1(0) = 0 , +1 k = 0,1,2,L. (7b) 当非线性恢复力? f (x)为 x的奇函数时，初始输入函数为更多还原

【Abstract】 Physical and mechanical oscillatory systems are often governed by nonlineardifferential equations. In many cases, it is possible to replace a nonlineardifferential equation by a corresponding linear differential equation thatapproximates the original nonlinear equation closely enough to give useful results.Often such linearization is not feasible and for this situation the original nonlineardifferential equation itself must be dealt with. The general theory and method ofdealing with linear differential equations are highly developed branches ofmathematics, whereas very little of a general nature is known about arbitrarynonlinear differential equations. In general, the study of nonlinear differentialequations is restricted to a variety of special classes of the equations and themethod of solution usually involves one or more of a limited number oftechniques to achieve analytical approximations to the solutions. The most common and most widely studied methods of all analyticalapproximation methods for nonlinear differential equations are the perturbationmethods. These methods involve the expansion of a solution to a differentialequation in a series in a small parameter. They include the L-P method, the KBM 7<WP=118>Abstractmethod and the Multi-time expansion. However, these methods apply to weaklynonlinear oscillations only. The method of harmonic balance is another procedurefor determining analytical approximations to the solutions of differentialequations by using a truncated Fourier series representation. An importantadvantage of the method is its applicability to nonlinear oscillatory problems forwhich the nonlinear terms are not “small”, i.e., no perturbation parameter needsto exist. In general, the success of harmonic balance method requires that thenonlinear restoring force ? f (x) is an odd function of x , where x represents thedisplacement measured from the stable equilibrium position. If this condition isnot satisfied, the method of harmonic balance, when used in lowest order, leads toinconsistencies. In addition, applying the method of harmonic balance or itsvarious generalization to construct higher-order approximate analytical solutionsis also very difficult, since they require analytical solution of sets of algebraicequation with very complex nonlinearity. In this dissertation, some analytical approximate methods are presented tosolve large amplitude nonlinear oscillations of single-degree-of-freedomconservative systems. The most interesting features of these new methods are itssimplicity and its excellent accuracy in a wide range of values of oscillationsamplitudes. These analytical approximate periods and corresponding periodicsolutions are valid for small as well as large amplitudes of oscillation, includingthe case of amplitude of oscillation tending to infinity.1. A modified method of harmonic balance for large amplitude nonlinear oscillations Consider a single-degree-of-freedom conservative system governed by 8<WP=119>Abstract d2x + f (x)= 0, (1a) dt2 x(0) = β , dx (0)= 0. (1b) dt By introducing an independent variableτ = ωt , and equation (1a,b) can berewritten as ω2x′′+ f (x)= 0, (2a) x(0) = β , x′(0) = 0 (2b)where a prime represents derivative with respect toτ . The new independentvariable is chosen in such a way that the solution to Eq.(2a,b), is a periodicfunction ofτ of period 2π . The corresponding period of the nonlinear oscillationis given byT = 2π ω . Here, both the periodic solution x(τ ) and frequencyωdepend on β . Let x(τ ) = x0(τ )+ ?x0(τ ), where x0(τ ) is an initial approx更多还原