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
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
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
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
Physical and mechanical oscillatory systems are often governed by nonlinear
differential equations. In many cases, it is possible to replace a nonlinear
differential equation by a corresponding linear differential equation that
approximates the original nonlinear equation closely enough to give useful results.
Often such linearization is not feasible and for this situation the original nonlinear
differential equation itself must be dealt with. The general theory and method of
dealing with linear differential equations are highly developed branches of
mathematics, whereas very little of a general nature is known about arbitrary
nonlinear differential equations. In general, the study of nonlinear differential
equations is restricted to a variety of special classes of the equations and the
method of solution usually involves one or more of a limited number of
techniques to achieve analytical approximations to the solutions.
The most common and most widely studied methods of all analytical
approximation methods for nonlinear differential equations are the perturbation
methods. These methods involve the expansion of a solution to a differential
equation in a series in a small parameter. They include the L-P method, the KBM
7
Abstract
method and the Multi-time expansion. However, these methods apply to weakly
nonlinear oscillations only. The method of harmonic balance is another procedure
for determining analytical approximations to the solutions of differential
equations by using a truncated Fourier series representation. An important
advantage of the method is its applicability to nonlinear oscillatory problems for
which the nonlinear terms are not “small”, i.e., no perturbation parameter needs
to exist. In general, the success of harmonic balance method requires that the
nonlinear restoring force ? f (x) is an odd function of x , where x represents the
displacement measured from the stable equilibrium position. If this condition is
not satisfied, the method of harmonic balance, when used in lowest order, leads to
inconsistencies. In addition, applying the method of harmonic balance or its
various generalization to construct higher-order approximate analytical solutions
is also very difficult, since they require analytical solution of sets of algebraic
equation with very complex nonlinearity.
In this dissertation, some analytical approximate methods are presented to
solve large amplitude nonlinear oscillations of single-degree-of-freedom
conservative systems. The most interesting features of these new methods are its
simplicity and its excellent accuracy in a wide range of values of oscillations
amplitudes. These analytical approximate periods and corresponding periodic
solutions are valid for small as well as large amplitudes of oscillation, including
the 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
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 be
rewritten as
ω2x′′+ f (x)= 0, (2a)
x(0) = β , x′(0) = 0 (2b)
where a prime represents derivative with respect toτ . The new independent
variable is chosen in such a way that the solution to Eq.(2a,b), is a periodic
function ofτ of period 2π . The corresponding period of the nonlinear oscillation
is given byT = 2π ω . Here, both the periodic solution x(τ ) and frequencyω
depend on β .
Let x(τ ) = x0(τ )+ ?x0(τ ), where x0(τ ) is an initial approx
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102