Duffing 系统的主-亚谐联合共振 1)

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李航^*^,^?, 申永军^,^*^,^?^,²⁾, 李向红^*^,^**, 韩彦军^?, 彭孟菲^?

* 石家庄铁道大学交通工程结构力学行为与系统安全国家重点实验室,石家庄 050043

? 石家庄铁道大学机械工程学院,石家庄 050043

** 石家庄铁道大学数理系,石家庄 050043

PRIMARY AND SUBHARMONIC SIMULTANEOUS RESONANCE OF DUFFING OSCILLATOR ¹⁾

Li Hang^*^,^?, Shen Yongjun^,^*^,^?^,²⁾, Li Xianghong^*^,^**, Han Yanjun^?, Peng Mengfei^?

* State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures, Shijiazhuang Tiedao University,Shijiazhuang 050043,China

? Department of Mechanical Engineering,Shijiazhuang Tiedao University,Shijiazhuang 050043,China

** Department of Mathematics and Physics,Shijiazhuang Tiedao University,Shijiazhuang 050043,China

通讯作者: 2)申永军,教授,主要研究方向：机械系统动力学与控制. E-mail:shenyongjun@126.com

收稿日期:2019-12-9接受日期:2020-02-20网络出版日期:2020-03-18

基金资助:

1)国家自然科学基金资助项目.U1934201
国家自然科学基金资助项目.11772206

Received:2019-12-9Accepted:2020-02-20Online:2020-03-18
作者简介 About authors

摘要
以Duffing系统为研究对象,研究在多频激励下同时发生主共振和1/3次亚谐共振的动力学行为与稳定性.首先,通过多尺度法得到系统的近似解析解,利用数值方法检验近似程度,结果吻合良好,证明了求解过程和解析解的正确性.然后,从解析解中导出稳态响应的幅频方程和相频方程,从幅频曲线以及相频曲线中发现系统最多存在7个不同的周期解,这种多解现象可用于对系统状态进行切换.基于Lyapunov稳定性理论,得到联合共振定常解的稳定条件,利用该条件分析了系统的稳定性,并与Duffing系统的主共振和1/3次亚谐共振单独存在时比较.最后,通过数值方法分析了非线性项和外激励对系统动力学行为与稳定性的影响,发现了联合共振特有的现象：刚度软化时,非线性项不仅影响系统的响应幅值,同时还影响系统的多值性和稳定性;刚度硬化时,非线性项对系统的影响与单一频率下主共振和1/3次亚谐共振类似,仅影响系统的响应幅值.这些结果对Duffing系统动力学特性的研究具有重要意义.
关键词： 非线性振动;非线性微分方程;Duffing系统;联合共振

Abstract
In this paper, the dynamics and stability of the Duffing oscillator subjected to the primary resonance together with the 1/3 subharmonic resonance are studied. At first, the approximate analytical solution and amplitude-frequency equation are obtained through the method of multiple scales, and the correctness and satisfactory precision of the approximate solution are verified by simulation. Then, the amplitude-frequency equation and phase-frequency equation of steady-state response are derived from the approximate analytical solution, and it can be found there are at most seven different periodic solutions, which are called multi-value characteristics and can be used to switch the state of the system. Moreover, the stability condition of steady-state response is derived based on Lyapunov theory, and the amplitude-frequency curves of steady-state response are compared with the cases where the primary or 1/3 subharmonic resonance exists alone, and it is found that the system contains both resonance characteristics. At last, the effects of nonlinear factor and excitations on the system response are analyzed by simulation. The particular phenomena in this system are revealed, i.e., the nonlinear factor affects the response amplitude, multi-value characteristics and stability of the system with stiffness softening. However, for the stiffness hardening system, the nonlinear factor only affects the response amplitude, which is similar to the cases of single-frequency excitation. These results are important for the study on the Duffing system or other similar systems.
Keywords：nonlinear vibration;nonlinear differential equation;Duffing oscillator;simultaneous resonance

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本文引用格式
李航, 申永军, 李向红, 韩彦军, 彭孟菲. Duffing 系统的主-亚谐联合共振 ¹⁾. 力学学报[J], 2020, 52(2): 514-521 DOI:10.6052/0459-1879-19-349
Li Hang, Shen Yongjun, Li Xianghong, Han Yanjun, Peng Mengfei. PRIMARY AND SUBHARMONIC SIMULTANEOUS RESONANCE OF DUFFING OSCILLATOR ¹⁾. Chinese Journal of Theoretical and Applied Mechanics[J], 2020, 52(2): 514-521 DOI:10.6052/0459-1879-19-349

引言

Duffing系统是动力学中一类典型的非线性系统,能够描述工程领域中的诸多非线性模型,例如转子系统的非线性行为^[1-3],船的横摇运动^[4-5],大型结构的振动^[6]等.在动力学领域,目前对Duffing系统的研究主要为周期振动解和混沌控制两方面.韩祥临等^[7]利用广义变分迭代方法研究了随机激励下Duffing系统的渐进解,并讨论了解的一致有效性.李瑞红等^[8]研究了一类含三次耦合项的二自由度Duffing系统,发现一种由周期运动直接通往混沌的途径.Shen等^[9-10]研究了一类含分数阶微分项的Duffing系统,提出等效刚度和等效阻尼的概念.Holmes等^[11]用二阶平均法研究了一类具有负非线性刚度的Duffing系统,分析了周期解的分岔行为.张毅等^[12]以多频参数激励Duffing系统为模型,基于快慢分析法得到模型的快子系统和慢变量,分析了快子系统的分岔行为.曲子芳等^[13]以周期变化的双频激励van derPol-Duffing系统为模型,研究了系统的簇发振荡模式及非光滑行为演化机制,给出了平衡曲线和分岔图及在非光滑边界产生非光滑行为的演化行为分析.吕小红和罗冠炜^[14]基于网格划分的思想设计了非线性系统多参数分岔的计算方法,利用此方法分析了Duffing系统在双参数平面上的分岔特性.毕勤胜和陈予恕^[15-16]研究了一类强非线性Duffing系统,利用功能关系得到系统的周期解,给出系统从主共振到1/3次亚谐分岔的转迁集,应用广义牛顿法得到系统的对称破缺分岔转迁集的解析表达式.Kimiaeifar等^[17]研究了一类van der Pol-Duffing系统,利用同伦分析法得到了系统的周期解.Jin和Hu^[18-19]研究了一类具有滞后状态反馈的Duffing系统在窄带随机参数激励下的主共振,和一类双时滞Duffing系统在窄带随机激励下的反馈控制,从振动控制的角度讨论了反馈增益和时滞对系统的影响.戎海武等^[20]研究了Duffing系统在谐和与窄带随机噪声联合激励下的参数主共振响应和稳定性问题,分析了系统的失稳和跳跃现象.Hosseini^[21]研究了Duffing系统的主共振,讨论了高阶近似解中的伪解问题,提出一种检测频率响应方程中是否存在伪解的判据.

以往对各类Duffing系统的周期振动解的研究可大致分为两类,一类是从系统结构角度,考虑结构的复杂性以建立更符合工程实际的动力学模型,例如文献^[9,10]中的分数阶微分项可以更好地描述系统中的黏弹性阻尼;另一类是研究复杂激励下系统的动力学行为,例如文献^[7,18,22]研究了随机激励下Duffing系统的解. Nayfeh在其专著^[23]中利用多尺度法给出了Duffing系统的3倍超谐与1/3次亚谐联合共振的解. 姜源等^[24-25]做了更进一步的工作,利用平均法得到了分数阶Duffing系统和van der Pol系统的3倍超谐与1/3次亚谐联合共振的解,并分析了分数阶项对系统动力学行为的影响.

在实际问题中,一个复杂的系统往往受到多个激励源同时作用.以汽车系统为例^[26],汽车在行进过程中的振动激励源主要是发动机激励、路面激励和风激励,这些激励通常含有不同的频率成分.Duffing系统在受多频激励时^[23-24]具有更加复杂的动力学现象,尤其是联合共振较为突出.目前对Duffing系统周期振动的研究主要是在单频激励下发生主共振^[9,21]、亚谐共振^[27-29]或超谐共振^[10,30-31],或者在多频激励下发生超谐-亚谐联合共振^[23-24],而对主-亚谐联合共振的研究尚未见报导.本文以多频激励的Duffing系统为对象,研究其同时发生主共振和1/3次亚谐共振时的动力学行为与稳定性.

1 Duffing系统主-亚谐联合共振的一次近似解

受多频激励的Duffing系统可以描述为

(1) $\begin{array}\ddot {u}\left( t \right) + 2\xi \omega _0 \dot {u}\left( t \right) + \omega _0^2 u\left( t \right) + \alpha _1 u^3\left( t \right) = \\ \qquad F_1 \cos \omega _1 t + F_2 \cos \omega_2 t \end{array}$

为研究系统的主-亚谐联合共振,对系统参数做如下限制 $ \omega _1 = \omega _0 + \varepsilon \sigma _1$, $\omega _2 = 3\omega _0 + \varepsilon \sigma _2$, $\xi \omega _0 = \varepsilon \mu$, $\alpha _1 = \varepsilon \alpha$, $ F_1 = \varepsilon f$, $f = O\left( 1 \right) $, $\sigma _1 = O\left( 1 \right) $, $\sigma _2 = O\left( 1 \right) $. 这样式(1)成为

(2)$ \begin{array}\ddot {u}\left( t \right) + \omega _0^2 u\left( t \right) = \varepsilon \left[ {f\cos {\kern 1pt} \left( {\omega _0 t + \varepsilon \sigma _1 t} \right) - 2\mu \dot {u}\left( t \right) - } \right. \\ \qquad \left. { \alpha u^3\left( t \right)} \right] + F_2 \cos \left( {3\omega _0 t + \varepsilon \sigma _2 t} \right) \end{array}$

应用多尺度法^[32]研究系统的一次近似解. 引入两个时间尺度$T_0 = t$和$T_1 =\varepsilon t$,并假设系统(2)的解具有如下形式

(3)$ u\left( t \right) = u_0 \left( {T_0 ,T_1 } \right) + \varepsilon u_1 \left({T_0 ,T_1 } \right)$

将式(3)代入式(2),比较$\varepsilon $的同次幂,得到一偏微分方程组

(4a)$D_0^2 u_0 + \omega _0^2 u_0 = F_2 \cos \left( {3\omega _0 T_0 + \sigma _2 T_1 } \right)$

(4b)$ D_0^2 u_1 + \omega _0^2 u_1 = - 2D_0 D_1 u_0 + f\cos \left( {\omega _0 T_0 + \sigma _1 T_1 } \right) - 2\mu D_0 u_0 - \alpha u_0^3 $

式(4a)的解为

(5) $\begin{array} \babl u_0 \left( {T_0 ,T_1 } \right) = a\left( {T_1 } \right)\cos \left[ {\omega _0 T_0 + \beta \left( {T_1 } \right)} \right] + \\ \qquad \dfrac{F_2 }{\omega _0^2 - \omega _2^2 }\cos \left( {3\omega _0 T_0 + \sigma _2 T_1 } \right) \end{array}$

也可以写成复数形式

(6) $ u_0 \left( {T_0 ,T_1 } \right) = A\left( {T_1 } \right){\rm e}^{{\rm j}\omega _0 T_0 } + B{\rm e}^{{\rm j}\left( {3\omega _0 T_0 + \sigma _2 T_1 } \right)} + cc $

其中$A\left( {T_1 } \right) = \dfrac{a\left( {T_1 } \right)}{2}{\rm e}^{{\rm j}\beta \left( {T_1 }\right)}$,$B = \dfrac{F_2 }{2\left( {\omega _0^2 - \omega _2^2 }\right)}$,$cc$为前述所有项的共轭,$a\left( {T_1 } \right)$ 和$\beta \left( {T_1 } \right)$分别为慢变振幅和相位.

将式(6)代入式(4b),为消除永年项,要求

(7)$ \begin{array}- 6\alpha AB^2 - 2{\rm j}\mu \omega _0 A - 3\alpha A^2\overline A - \\ \qquad 3\alpha \bar {A}^2B{\rm e}^{{\rm j}\sigma _2 T_1 } +\dfrac{f}{2}{\rm e}^{{\rm j}\sigma _1 T_1 } - 2{\rm j}\omega _0 D_1 A = 0 \end{array}$

分离式(7)的实部和虚部,得到慢变振幅$a$和相位$\beta $满足的微分方程组

(8a)$ D_1 a = - \mu a + \dfrac{3\alpha a^2B}{4\omega _0 }\sin \left( {3\beta - \sigma _2 T_1 } \right) - \dfrac{f}{2\omega _0 }\sin \left( {\beta - \sigma _1 T_1 } \right) $

$ D_1 a = - \mu a + \dfrac{3\alpha a^2B}{4\omega _0 }\sin \left( {3\beta - \sigma _2 T_1 } \right) - \dfrac{f}{2\omega _0 }\sin \left( {\beta - \sigma _1 T_1 } \right) $ (8a)

(8b)$ aD_1 \beta = \dfrac{3\alpha aB^2}{\omega _0 } + \dfrac{3\alpha a^3}{8\omega _0 } + \dfrac{3\alpha a^2B}{4\omega _0 }\cos \left( {3\beta - \sigma _2 T_1 } \right) - \dfrac{f}{2\omega _0 }\cos \left( {\beta - \sigma _1 T_1 } \right) $

从而系统(2)的一次近似解可以表示为

(9)$ u_0 \left( t \right) = a\cos \left( {\omega _0 t + \beta } \right) + 2B\cos \omega _2 t $

其中,$a$和$\beta $由式(8)确定.

2 定常解及其稳定条件

从式(8)可以看出,系统定常解存在的必要条件是$\beta - \sigma _1 T_1 $和$3\beta - \sigma _2 T_1$均为常数,这时有$D_1 \beta = \sigma _1 = \sigma _2 / 3$,进而有$\omega _1 = \omega _2 /3$,即只有当两个激励频率满足特定倍数关系时,才能求得主-亚谐联合共振的定常解. 设$\sigma = \sigma _1 =\sigma _2 / 3$,$\beta - \sigma T_1 = \varphi $,从而式(8)可以写成自治微分方程组

(10a)$ D_1 a = - \mu a + \dfrac{3\alpha a^2B}{4\omega _0 }\sin 3\varphi - \dfrac{f}{2\omega _0 }\sin \varphi $

(10b)$ aD_1 \varphi = - \sigma a + \dfrac{3\alpha aB^2}{\omega _0 } + \dfrac{3\alpha a^3}{8\omega _0 } + \dfrac{3\alpha a^2B}{4\omega _0 }\cos 3\varphi - \dfrac{f}{2\omega _0 }\cos \varphi $

相应的一次近似解成为

(11)$ u\left( t \right) = a\cos \left( {\omega_1 t + \varphi } \right) + 2B\cos \omega_2 t $

为检验式(10)和式(11)确定的一次近似解的精确程度,取一组参数$\mu =0.1$,$\varepsilon = 0.1$,$\alpha = 1$,$\omega _0 = 1$,$F_1 = 0.2$,$F_2 = 2$,计算稳态响应的幅频特性,计算时间$t = 1000$ s,将前800 s响应略去,取后200 s响应的幅值为稳态幅值,得到系统(2)的幅频响应曲线如图1所示.

图1

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图1幅频曲线的比较

Fig. 1Comparison of amplitude-frequency responses

取激励频率$\omega _1 = 1.22$, $\omega _2 = 3\omega _1 $, 初值$\left( {a_0 , \varphi _0 }\right) = \left( {0.1, 0} \right)$, 代入式(10)计算$\left( {a, \varphi }\right)$,再将结果代入式(11)计算近似解;将式(11)求导得到速度响应$\dot {u}\left( t\right)$,再将$\left( {a_0 , \varphi _0 } \right)$代入式(11)和$\dot {u}\left( t\right)$,得到$(u_0 ,\dot {u}_0 ) = $$( - 0.06,0)$,然后将其作为初值代入系统(2)计算数值解,最后得到系统(2)的位移时间历程如图2所示.图1和图2中,圆圈表示数值解,实线表示解析解. 可见,解析解的近似程度良好.

图2

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图2位移时间历程的比较

Fig. 2Comparison of displacement time histories

从幅频曲线中可以看到,系统在一定频率范围内存在多解现象.事实上,若调整仿真的初始条件可以得到更多的解,而系统多解和不稳定的现象受近似解中的第一部分,即$a\cos(\omega _1 t + \varphi )$支配. 因此,在系统定常解的稳定性分析中,只需考察这一部分.

令式(10)中$D_1 a = 0$, $D_1 \varphi = 0$, 可以得到稳态振幅$\bar {a}$和相位$\bar {\varphi}$满足的代数方程组

(12a)$\mu \bar {a} = \dfrac{3\alpha \bar {a}^2B}{4\omega _0 }\sin 3\bar {\varphi } - \dfrac{f}{2\omega _0 }\sin \bar {\varphi } $

(12b)$\sigma \bar {a} - \dfrac{3\alpha \bar {a}B^2}{\omega _0 } - \dfrac{3\alpha \bar {a}^3}{8\omega _0 } = \dfrac{3\alpha \bar {a}^2B}{4\omega _0 }\cos 3\bar {\varphi } - \dfrac{f}{2\omega _0 }\cos \bar {\varphi } $

进一步可以得到幅频响应方程和相频响应方程分别为

(13a)$ \left[ { - 48\alpha \sigma \omega _0 \left( {\bar {a}^2 + 8B^2} \right)} \right. + 3\alpha \left( {3\alpha \bar {a}^4 + 36\alpha \bar {a}^2B^2 + } \right. \\ \qquad \left. {\left. {192 \alpha B^4 - 8fB} \right) + 64\omega _0^2 \left( {\mu ^2 + \sigma ^2} \right)} \right]^2 \cdot \\ \qquad \left[ {9\alpha ^2\bar {a}^6 + } \right.108\alpha ^2B^2\bar {a}^4 + 48\alpha B\bar {a}^2\left( {12\alpha B^3 + f} \right) - \\ \qquad \left. { 48\sigma \alpha \omega _0 \bar {a}^2\left( {\bar {a}^2 + 8B^2} \right) + 64\bar {a}^2\omega _0^2 \left( {\mu ^2 + \sigma ^2} \right) - 16f^2} \right] = \\ \qquad 98 304\alpha B\mu ^2\omega _0^2 f^3 $

(13b)$ 16f^2 - 9\bar {a}^6\alpha ^2 - 108\bar {a}^4\alpha ^2B^2 - 576\bar {a}^2\alpha ^2B^4 + \\ \qquad 48\sigma \bar {a}^2\alpha \omega _0 \left( {\bar {a}^2 + 8B^2} \right) - 64\bar {a}^2\omega _0^2 \left( {\sigma ^2 + \mu ^2} \right) = \\ \qquad 48\bar {a}^2\alpha Bf\cos 2\bar {\varphi } $

下面考察稳态解的稳定性,用慢变振幅$a$和相位$\varphi $组成二维状态向量${\pmb V} = [a,\varphi ]^{\rm T}$,构造二维向量函数

(14)$\begin{array} {\pmb F}\left( {\pmb V} \right) = \left[ \!\! \begin{array}{c} {f_1 \left( {a,\varphi } \right)} \\ {f_2 \left( {a,\varphi } \right)} \end{array} \!\! \right] = \\ \left[ \!\! \begin{array}{c} { - \mu a + \dfrac{3\alpha a^2B}{4\omega _0 }\sin 3\varphi - \dfrac{f}{2\omega _0 }\sin \varphi } \\ { - \sigma + \dfrac{3\alpha B^2}{\omega _0 } + \dfrac{3\alpha a^2}{8\omega _0 } + \dfrac{3\alpha aB}{4\omega _0 }\cos 3\varphi - \dfrac{f}{2a\omega _0 }\cos \varphi } \end{array}\!\! \right] \end{array}$

在稳态解$(\bar {a},\bar {\varphi })$处,向量函数${\pmb F}({\pmb V})$的Jacobi矩阵和特征方程分别为

(15a)${\pmb J} = \left[\!\!\begin{array}{cc} {\dfrac{3\bar {a}\alpha B\sin 3\bar {\varphi }}{2\omega _0 } - \mu } & {\dfrac{9\alpha \bar {a}^2B\cos 3\bar {\varphi }}{4\omega _0 } - \dfrac{f\cos \bar {\varphi }}{2\omega _0 }} \\ {\dfrac{3\bar {a}\alpha }{4\omega _0 } + \dfrac{3\alpha B\cos 3\bar {\varphi }}{4\omega _0 } + \dfrac{f\cos \bar {\varphi }}{2\bar {a}^2\omega _0 }} \!&\! {\dfrac{f\sin \bar {\varphi }}{2\bar {a}\omega _0 } - \dfrac{9\alpha \bar {a}B\sin 3\bar {\varphi }}{4\omega _0 }} \end{array}\!\! \right]$

(15b)$\lambda ^2 - P\lambda + Q = 0 $

其中$P = {\rm tr}{\pmb J} $,$Q = \det [{\pmb J} ]$.

由Lyapunov稳定性理论可知,稳态解渐进稳定的条件是$P < 0$且$Q > 0$. 对于阻尼系统,恒有$P <0$.因而Duffing系统主-亚谐联合共振的定常解稳定条件为

(16)$ 27\bar {a}^{6}\alpha ^2 - {1728}\bar {a}^{2}\alpha ^2B^{4} + {48}\bar {a}\alpha f\cos \bar {\varphi }\left( {\bar {a}^2 + 16B^2} \right) - \\ \qquad 192\bar {a}^2\omega _0^2 \left( {\sigma ^2 + \mu ^2} \right) - 64f^2 + 128\bar {a}\omega _{0} \left( {9\sigma \bar {a}\alpha B^2 - } \right. \\ \qquad \left. { 2f \mu \sin \bar {\varphi } - 2f\sigma \cos \bar {\varphi }} \right) > 0$

取一组参数$\mu = 0.1$,$\varepsilon = 0.1$,$\alpha = 1$,$\omega _0 = 1$,$F_1 = 0.1$,$F_2=4$,利用式(13)计算稳态响应的幅频特性和相频特性,式(16)判断稳定性,得到幅频曲线和相频曲线分别如图3和图4所示,其中圆圈表示稳定解,星号表示不稳定解.从图3和图4可见,第一部分最多存在7个解s1$\sim$s7,所以系统(2)也最多存在7个解S1$\sim $S7,其中有4个稳定解,3个不稳定解. 以$\sigma =2$为例,系统(2)的3个稳定解S1,S2和S4的周期轨道如图5所示.将图3分别与主共振和1/3次亚谐共振^[33]比较可以看出,s1$\sim $s3是主共振的特性,s4$\sim$s7是1/3次亚谐共振的特性.

图3

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图3定常解的幅频响应

Fig. 3Amplitude-frequency curves of steady-state response

图4

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图4定常解的相频响应

Fig. 4Phase-frequency curves of steady-state response

图5

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图5系统的周期轨道

Fig. 5Periodic orbits

3 系统参数的影响

为确定非线性系数$\alpha $对系统响应的影响,固定一组参数$\mu = 0.1$,$\varepsilon = 0.1$,$\omega _0 = 1$,$F_1 = 0.1$,$F_2 = 8$并改变$\alpha $进行数值计算,其中,调谐参数$\sigma $的范围取$\left[ { - 5,5} \right]$,也即$\omega _1 $的范围为$\left[ {0.5,1.5} \right]$,$\omega _2 $的范围为$\left[ {1.5,4.5} \right]$. 图6给出非线性系数$\alpha $对系统的影响. 结果显示,当$\alpha > 0$时,在一定频率范围内主要影响振幅,刚度逐渐硬化使得响应振幅减小,幅频曲线的弯曲程度增加,改变了系统的频率特性;当$\alpha < 0$时,对振幅、多值性和稳定性均有影响,刚度软化使得响应的共振峰逐渐减小,趋于稳定.

此外,为确定两个激励幅值对系统响应的影响,固定参数$\mu = 0.1$,$\varepsilon = 0.1$,$\omega _0 = 1$,$\alpha = 1$,$F_2 = 16$改变$F_1 $、固定参数$\mu = 0.1$,$\varepsilon = 0.1$,$\omega _0 = 1$,$\alpha= 1$,$F_1 = 0.1$改变$F_2 $,计算得到系统的幅频曲线分别如图7和图8所示. 从图中可以看到$F_1$对幅频特性的骨架线影响很小,对幅频曲线的形态影响较大;而$F_2$增大使联合共振的骨架线向高频附近移动,幅频曲线的形态变化不大.

图6

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图6非线性系数$\alpha $的影响

Fig. 6Effects of the nonlinear coefficient $\alpha $

图7

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图7$F_1 $的影响

Fig. 7Effects of excitation amplitude $F_1 $

图8

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图8$F_2 $的影响

Fig. 8Effects of excitation amplitude $F_2 $

4 结论

本文利用多尺度法得到了Duffing系统的主-1/3次亚谐联合共振的解析解.由Lyapunov稳定性理论得到了定常解的稳定条件.基于此条件分析系统响应的稳定性,发现系统响应既有主共振的特性又有1/3次亚谐共振的特性,各个解支的稳定性与仅发生主共振或亚谐共振时相同.讨论了系统参数对定常解的幅频特性和稳定性的影响,发现在一定频率范围内,非线性系数$\alpha$分别取正负时对系统响应有截然不同的影响：$\alpha > 0$时,仅影响各个解支的幅值,$\alpha <0$时,对解的数量、稳定性和幅值均有影响. 此外,外激励的幅值$F_1 $和$F_2$也分别影响着幅频曲线的形态和骨架线.

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